What is Differentiability: Definition and 196 Discussions

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp.
More generally, for x0 as an interior point in the domain of a function f, then f is said to be differentiable at x0 if and only if the derivative f ′(x0) exists. In other words, the graph of f has a non-vertical tangent line at the point (x0, f(x0)). The function f is also called locally linear at x0 as it is well approximated by a linear function near this point.

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  1. W

    Continuity and differentiability in two variables

    Hi If the function ##f(x,y)## is independently continuous in ##x## and ##y##, i.e. f(x+d_x,y) = f(x,y) + \Delta_xd_x + O(d_x^2) and f(x,y+d_y) = f(x,y) + \Delta_yd_y + O(d_y^2) for some finite ##\Delta_x##, ##\Delta_y##, and small ##\delta_x##, ##\delta_x##, does it mean that it is continuous...
  2. V

    Allowed values for the "differentiability limit" in complex analysis

    In complex analysis differentiability for a function ##f## at a point ##z_0## in the interior of the domain of ##f## is defined as the existence of the limit $$ \lim_{h\rightarrow{}0}\frac{f(z_0+h)-f(z_0)}{h}.$$ But why are the possible ##z_0##'s in the closure of the domain of the original...
  3. U

    Definition & Proving Differentiability: A Function f at a Point a

    (a) State precisely the definition of: a function f is differentiable at a ∈ R. (b) Prove that, if f is differentiable at a, then f is continuous at a. You may assume that f '(a) = lim {f(x) - f(a)}/(x - a) x→a (c) Assume that a function f is differentiable at each x∈ R and...
  4. J

    On differentiability on endpoints of an open interval

    Before asking a question I would first like to mention the definitions of limit of function and differentiality at x=p 1) Limit of function (f) at x=p Let E be domain of f and p be a limit point of E. Let Y be the range of f. If there exists q∈E such that for all ε>0 there exists δ>0...
  5. D

    Two varibale function. Continuity, derivability and differentiability

    Homework Statement Discuss the continuity, derivability and differentiability of the function f(x,y) = \frac{x^3}{x^2+y^2} if (x,y)≠(0,0) and 0 otherwise Homework Equations if f is differentiable then ∇f.v=\frac{∂f}{∂v} if f has both continuous partial derivative in a neighbourhood of x_0...
  6. A

    MHB Functions with asymptotes and differentiability Question

    Hey guys, More questions for you guys this time, these seem easy but always have a few nuances I seem to miss. With that said, I'd greatly appreciate your guys' help. Question: For 1a, I sketched two straight lines where x=/ 1 and y=1/2 for one line and y= -1/2 for the other. Thus, x=1 is...
  7. binbagsss

    Differentiability and continously differentiable definition/concepts.

    Theorem: ctsly differentiable at a if the function is cts and its partial derivatives exist and are cts in a neighborhood of a. [1] - so to be differentiable we can check whether this conditions holds, and if it does ctsly diff => diff. - the definition of a scalar function being...
  8. V

    Differentiability of piecewise multivariable functions

    I guess my first questions is whether saying that a function is differentiable is the same as saying that its derivative is continuous. i.e. if \lim_{x\rightarrow{}a}f'(x)=f'(a) then the function is differentiable at ##a##. Or is it just a matter of the value ##f'(a)## existing? Now my...
  9. Barioth

    MHB Is f(x,y) Differentiable at (0,0)?

    Hi, I'm having trouble évaluation the differentiability at (0,0) of the function f(x,y)=\frac{x^3}{x^2+y^2} for (x,y) not nul, and f(x,y)=0 if (x,y)=0 I know f is differentiable if (x,y) isn't nul since the partial derivative are continuous, but I don't know how to evaluate it at (0,0)...
  10. S

    MHB Differentiability of complex function

    I have found a question Prove that f(z)=Re(z) is not differentiable at any point. According to me f(z)=Re(z)=Re(x+iy)=x which is differentiable everywhere. Then where is the mistake?
  11. L

    Continuity and differentiability of a piecewise function

    Homework Statement Discuss the continuity and differentiability of f(x) = \begin{cases} x^2 & \text{if } x\in \mathbb{Q} \\ x^4 & \text{if } x\in \mathbb{R}\setminus \mathbb{Q} \end{cases} Homework Equations The Attempt at a Solution From the graph of ##f##, I can see...
  12. F

    Differentiability of a function

    Homework Statement I have the function f, defined as follows: f=0 if xy=0 f= ##xysin(\frac{1}{xy})## if ##xy \neq 0## Study the differentiability of this function. The Attempt at a Solution there are no problems in differentiating the function where ##xy\neq0##. the partials in (0,0)...
  13. M

    Problem on differentiability

    Homework Statement . Let ##f:\mathbb ℝ:→ℝ^2## be a function defined as: ##f(x,y)=\frac{x^2y-2xy+y} {(x-1)^2+y^2} \forall (x,y)≠(1,0)## and ##f(1,0)=0##. Prove that for any curve ##α:(-ε,ε)→ℝ^2## of class ##C^1## (where ##ε>0##) such that ##α (0)=(1,0)## and ##α(t)≠(1,0)## for every ##t≠0##, the...
  14. Q

    Differentiability using limit definition

    Homework Statement http://i.minus.com/jbzvT5rTWybpEZ.png Homework Equations If a function is differentiable, the function is continuous. The contrapositive is also true. If a function is not continuous, then it is not differentiable. A function is differentiable when the limit definition...
  15. S

    MHB Differentiability of a Complex Function

    f:\mathbb{C}\rightarrow\mathbb{C} \\ f(z)=\left\{\begin{array} \frac{(\bar{z})^2}/ {z} \quad z\neq0 \\ 0 \quad z=0 \end{array} \right. Show that f is differentiable at z=0, but the Cauchy Riemann Equations hold at z=0. Well i have tried to start the first part but i am stuck, could you...
  16. R

    Continuity and Differentiability

    Homework Statement f(x) = sin ∏x/(x - 1) + a for x ≤ 1 f(x) = 2∏ for x = 1 f(x) = 1 + cos ∏x/∏(1 - x)2 for x>1 is continuous at x = 1. Find a and b Homework Equations For a lim x→0 sinx/x = 1. The Attempt at a Solution I tried...
  17. J

    Strong differentiability condition

    Assume that a point x is an interior point of domain of some function f:[a,b]\to\mathbb{R}, and assume that the limit \lim_{(\delta_1,\delta_2)\to (0,0)} \frac{f(x+\delta_2)-f(x+\delta_1)}{\delta_2-\delta_1} exists. What does this imply? Well I know it implies that f'(x) exists, but...
  18. T

    Defining the continuity and differentiability of multi variate functio

    Let f: R2-->R be defined by f(x,y) = xy2/(x2+y2 if (x,y) ≠ 0, f(0,0) = 0 a) is f continuous on R2? b) is f differentiable on R2? c) Show that all the dirctional derivatives of f at (0.0 exist and compute them Attempt: a) I had an idea to show that multivariate functions are...
  19. G

    Rigorously determining differentiability in multiple variables

    Homework Statement Determine if f(x,y) = ((x-y)4 +x3 +xy2)/(x2+y2) [f(x,y = 0 @ (0,0)] is differentiable at the origin. Homework Equations x = (0,0) The Attempt at a Solution A function is differentiable at x if f(x+Δx) - f(x) = AΔx + |Δx|R(x) Where A are constant...
  20. S

    Rolle's theorem -> Differentiability

    Homework Statement So I'm doing problems where I have to verify Rolle's hypotheses. I am only having trouble with the differentiability part. My professor wants me to prove this. So for example, f(x)=√(x)-(1/3)x [0,9] Homework Equations none The Attempt at a Solution 1.) I know the...
  21. A

    Differentiability of Monotone Function's: Lebesgue's Theorem

    a)Does convergence imply being properly defined? So would it not be properly defined if it was divergent? b)I am having trouble why the last part (in the attachment) says, "Then, by (1), f(x_0) - f(x) \geq dfrac{2^k} for all [itex]x < x_0." But does (1) tell us that it's "equal" instead of...
  22. C

    Differentiability of a multivariable function?

    As a preface, this question is taken from Vector Calculus 4th Edition by Susan Jane Colley, section 2.3 exercises. Homework Statement "Explain why each of the functions given in Exercises 34-36 is differentiable at every point in its domain." 34. xy - 7x^8y^2 cosx 35. \frac{x + y +...
  23. Y

    MHB Solving Problems Involving Differentiability of a Function

    Hello, How do I solve this kind of problems ? For which values of x the next function is "differentiable" ? I know it has something to do with the existent of the one sided limits, but which limits should I be calculating exactly ? Thanks !
  24. P

    Leibniz's rule and differentiability of a function.

    Hi, Homework Statement (I) The following function is defined for α,β>0: f(x) = { xβsin(1/xα), x≠0; { 0, x=0 I was asked for the values of α,β for which f(x) would be continuous at 0, differentiable at 0, continuously differentiable at 0, and twice differentiable at 0. (II) I was asked to...
  25. L

    Differentiability of a Series of Functions

    I'm working on a problem where I need to show that the series of functions, f(x) = Ʃ (xn)/n2, where n≥1, converges to some f(x), and that f(x) is continuous, differentiable, and integrable on [-1,1]. I know how to show that f(x) is continuous, since each fn(x) is continuous, and I fn(x)...
  26. S

    Differentiability of absolute value

    f: R2 to R1 given by f(x,y) = x(|y|^(1/2)) show differentiable at (0,0) so I'm using the definition lim |h| ->0 (f((0,0) + 9(h1,h2)) - f(0,0) - Df(0,0) (h1,h2)) / |h| so first for the jacobian for f, when I'm doing the partial with respect to y, do I have to break this into the case y>0...
  27. B

    Continuity and differentiability over a closed interval

    Homework Statement http://i.imgur.com/69BmR.jpg Homework Equations The Attempt at a Solution a, c are right because f(c) is continuous. b, d are right because f'(c) is differentiable over the interval I am not sure about e. Can anyone explain to me?
  28. H

    What is the difference between differentiability and continuity at a point?

    Could someone explain this to me in terms of limits and derivatives instead of plain english? For example, how would you solve a question that says find whether the function f is differentiable at x=n and a question that asks find whether the function f is continuous at = n...
  29. D

    Differentiability of a multi-variable function

    For a function of a single variable, I can check if the function is differentiable by simply taking the limit definition of a derivative and if the limit exists, then the function is differentiable at that point. Differentiability also implies continuity at this level.Now, for a function of...
  30. estro

    Differentiability of a function

    Hello, I'm having problems figuring out theoretical problem on "differentiability of a function". [I hope that I spelled it right...] Suppose that: 1. Functions f(x,y) and g(x,y) are well defined in some little domain around (0,0). (*1) 2. g(x,y) is continuous at (0,0). (*2) 3. f(x,y)...
  31. B

    Differentiability and Continuity at a point

    Homework Statement Refer to attached file. The attempt at a solution (a) g'(0) = \lim_{x\rightarrow 0} {\frac{g(x)-g(0)}{x-0}} g'(0) = \lim_{x\rightarrow 0} {\frac{x^\alpha cos(1/x^2)-0}{x}} g'(0) = \lim_{x\rightarrow 0} {x^{(\alpha-1)} cos(1/x^2)-0} g'(0) = 0 So...
  32. R

    Differentiability of composite functions

    Hi, I have a small question about this. Using the chain rule, I know that a composition of differentiable functions is differentiable. But is it also true that if a composition of functions is differentiable, then all the functions in the composition must be differentiable? For example, if...
  33. N

    Complex Analysis - Differentiability

    Homework Statement Show that the function f defined by f(z) = 3\,{x}^{2}y+{y}^{3}-6\,{y}^{2}+i \left( 2\,{y}^{3}+6\,{y}^{2}+9\,x \right) is nowhere differentiable.The Attempt at a Solution Computing the C.R equations for this, I am left with {y}^{2}+2\,y={\it xy} and x^2+(y-2)^2 = 1...
  34. T

    Functional differentiability: Frechet, but not Hadamard?

    I have a question regarding functional differentiablility. I understand that Frechet differntiability of a functional T with respect to a norm \rho_1 implies Hadamard differentiability of the functional T with respect to the same norm. However, it is no surprise that there would be cases...
  35. M

    Proof of a limit involving definition of differentiability

    Homework Statement let the function f:ℝ→ℝ be differentiable at x=0. Prove that lim x→0 [f(x2)-f(0)] ______________ =0 x Homework Equations The Attempt at a Solution I am kind of lost on this one, I have tried manipulating the definition of a...
  36. N

    Is my methodology for checking differentiability and analyticity correct?

    Homework Statement State the Cauchy-Riemann equations and use them to show that the function defined by f(z) = |z|^2 is differentiable only at z = 0. Find f′(0). Where is f analytic? The Attempt at a Solution f(z) = |z|^2 = (x^2 + y^2) \frac{du}{dx} = 2x, \frac{dv}{dy} = 0 \frac{du}{dy} =...
  37. M

    Understanding Differentiability of f at x = 0

    The paragraph says, " Even if the function f is an everywhere differentiable function, it is still possible for f ' to be discountinuous. However, the graph of f ' can never exhibit a discountinuity of ..." picture is in paint document... What type of discountinuity is that? a hole...
  38. S

    Analytic proof of continuity, differentiability of trig. functions

    Since I am new to PF (hi!), before I go any further, I would like to a) briefly note that this is an independent study question, and that its scope goes beyond that of a textbook question - i.e., I believe that this thread belongs here - and b) also note that I am new to analysis and early...
  39. R

    Continuous and smooth on a compact set implies differentiability at a point

    I'm trying to prove that if a function is continuous on [a,b] and smooth on (a,b) then there's a point x in (a,b) where f'(x) exists. The definition of smoothness is: f is smooth at x iff \lim_{h \rightarrow 0} \frac{ f(x+h) + f(x-h) - 2f(x) }{ h } = 0 . I'm starting with the simpler case...
  40. J

    What Are the Derivatives of |x| and [x]?

    Are |x| and [x] differentiable anywhere? If so, what're their derivatives?
  41. R

    A problem about differentiability

    as you know i have been asked a question which no no way i couldn't tackle it. and its is about differentiabilty. at long last i found a solution. i want to share with you. could you check out please. thanks for now.this is the question. and this is my solution.(i assume that when x goes...
  42. R

    A problem about differentiability

    i tried to solve this problem. i can do it a little. but i can't progress. as far as I'm concerned, it requires outstanding performance. thanks for now... PROBLEM MY SOLUTION...
  43. R

    A problem about differentiability

    i need your helps. thanks for everything.
  44. P

    Better Understanding of Complex Differentiability

    So the way I understand complex differentiability and its requirement that the partial derivatives satisfy the Cauchy-Riemann Equations is that we would really like ℂ to have the same nice property as ℝ, that is to say we would really like the derivative to be a linear operator which is itself...
  45. Shackleford

    Analysis: Limits, strictly increasing, differentiability

    I've worked all of these out. I'm mostly confident I did them correctly, but I'm prone to overlook subtleties or counterexamples sometimes. http://i111.photobucket.com/albums/n149/camarolt4z28/1ab-1.png http://i111.photobucket.com/albums/n149/camarolt4z28/1gf2-1.png
  46. W

    Simple Differentiability and Continuity Question

    Homework Statement If f(x) = 3 for x < 0 and f(x) = 2x for x ≥ 0, is f(x) differentiable at x = 0? State and justify why/why not. Homework Equations The Attempt at a Solution Obviously, since f(x) is not continuous and the limit doesn't exist as x\rightarrow0, the function...
  47. U

    Proving Differentiability of a Piece-wise Function

    1. Suppose f(x)=0 if x is irrational, and f(x)=x if x is rational. Is f differentiable at x=0? 2. the derivative= lim[h->0] [f(a+h)-f(a)]/h 3. I don't really know how to start, but I do know that between any two real numbers, there exists a rational and irrational number. So I'm...
  48. S

    Is the Function B(x)= xsin(1/x) Differentiable at x=0?

    Homework Statement B(x)= xsin(1/x) when x is not equal to 0 = 0 when x is equal to 0 Determine if the function is differentiable at 0 Homework Equations The Attempt at a Solution I get B'(x)= sin(1/x)+cos(1/x)*(-1/x) but really do not know what should be done next...
  49. M

    Differentiability of a two variable function with parameter

    Homework Statement For which parameter \alpha\in\mathbb{R} the function: f(x, y)= \begin{cases}|x|^\alpha \sin(y),&\mbox{ if } x\ne 0;\\ 0, & \mbox{ if } x=0\end{cases} is differentiable at the point (0, 0)? The Attempt at a Solution For α<0, the function is not continue at (0, 0)...
  50. M

    A few problems and continuity and differentiability

    Homework Statement I have an upcoming math test, and these are from the sample exam. I'll post my solutions as I go along. I've submitted this post as is and am going to edit in my attempts. A few of these are "verify my proof is rigorous" others are "i've no idea what I'm doing 1 Using simple...
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