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. H

    Fourier series convergence - holder continuity and differentiability

    Homework Statement Given each of the functions f below, describe the set of points at which the Fourier series converges to f. b) f(x) = abs(sqrt(x)) for x on [-pi, pi] with f(x+2pi)=f(x) Homework Equations Theorem: If f(x) is absolutely integrable, then its Fourier series converges to f...
  2. A

    Some questions about differentiability

    We have a corollary that But I wonder can we prove a function is not differentiable by showing that f_{x} or f_{y} are not continuous? i.e. is the converse of this statement true? By the way, are there any books have a proof on this corollary? Most of the Calculus book state the...
  3. N

    Differentiability and extreme points question

    2.b) f is continues in [0,1] and differentiable in (0,1) f(0)=0 and for x\in(0,1) |f'(x)|<=|f(x)| and 0<a<1 prove: (i)the set {|f(x)| : 0<=x<=a} has maximum (ii)for every x\in(0,a] this innequality holds \frac{f(x)}{x}\leq max{|f(x)|:0<=x<=a} (iii)f(x)=0 for x\in[0,a] (iii)f(x)=0 for...
  4. 0

    Question about the differentiability of a function of more than one variable

    I've been thinking about this for a while... sorta. If a function of two or more variables is differentiable at some point, does this imply that all its partial derivatives are continuous at that point?
  5. F

    This problem is making me think, deeply about continuity and differentiability

    Homework Statement differentiability is a tough word to spell. F(x,y) = (x^2 + y^3)^{\frac{1}{3}} Find F_y (0,0) The Attempt at a Solutionhttp://www.wolframalpha.com/input/?i=D[%28x^2+%2By^3%29^%281%2F3%29%2Cy] But I get 0/0 I found the answer to be F_y (0,0) = \frac{\mathrm{d}...
  6. M

    Differentiability of BitXor function

    in many programming languages there is a function of two variables called BitXor (which is also known as nim-sum, since it is used in solving de nim game) which represents each number as a string of its binary digits and then takes the Xor of each pair of terms, forming a new number. For...
  7. S

    True or false; differentiability

    Homework Statement if g:[-1,1] -> Reals is differentiable with g(0) = 0 and g(x) doesn't equal 0 for x not = 0 and f : Reals -> Reals is a continuous function with f(x)/g(x) ->1 as x->0 then f(x) is differentiable at 0. Homework Equations The Attempt at a Solution I took...
  8. Y

    Differentiability of functions defined on manifolds

    Quoted from a book I'm reading: if f is any function defined on a manifold M with values in Banach space, then f is differentiable if and only if it is differentiable as a map of manifolds. what does it mean by 'differentiable as a map of manifolds'?
  9. L

    Differentiability of eigenvalues of a positive matrix

    I have a matrix A, which contains only positive real elements. A is a differentiable function of t. Are the eigenvalues of A differentiable by t?
  10. J

    True or false: Differentiability with vectors

    If all the first partial derivatives of f exist at \vec{x}, and if \lim_{\vec{h}\rightarrow\vec{0}}\frac {f(\vec{x})-(\nabla f(\vec{x}))\cdot\vec{h}}{||\vec{h}||} = 0 Then f is differentiable at \vec{x} Note: Its the magnitude of h on the bottom. First of all, I don't...
  11. O

    Differentiability of a Twice Differentiable Function

    Homework Statement Let g:R->R be a twice differentiable function satisfying g(0)=g'(0)=1 and g''(x)=g(x)=0 for all x in R. (i) Prove g has derivatives of all orders. (ii)Let x>0. Show that there exists a constant M>0 such that |g^n(Ax)|<=M for all n in N and A in (0,1). Homework Equations...
  12. B

    Differentiability and differential of a funtion

    For months I have been staring into this expression, and I cannot visualize what the hell omega represents... f(x)-f(x0)=f'(x0)(x-x0)+\omega(x)*(x-x0) Where \omega(x)(=\omega(x;\Deltax)) is a continuous function in point x0 and equals zero in that point or lim, as x approaches x0 of...
  13. L

    Is a Function Differentiable if Its Symmetric Derivative Exists?

    Homework Statement If a function satisfies g'(x) = lim(h->0) {[g(x+h)-g(x-h)/2h}, must g be differentiable at x? Provide a proof or counter example Homework Equations From the formal definition of differentiation, I know that g'(x) = lim (h->0) {[g(x+h)-g(x)]/h} The Attempt at a...
  14. L

    Identifying Non-Differentiable Points Without Graphing

    Homework Statement Differentiability- Okay, so I understand that a function is not differentiable if there are either: A. A cusp B.A jump C. f(x) DNE D. Vertical tangent E. Pretty much if there isn't a limit there is no derivative which means its not differentiable. How would one find the...
  15. Q

    Proving differentiability in two dimensions

    Homework Statement proof at 0,0 g(x,y) is differentiable Homework Equations notes says i have to write in the form fx(0,0)\Deltax + fy(0,0)\Deltay + E1\Deltax + E2\DeltayThe Attempt at a Solution i compute fx(0,0) = 0 and fy(0,0) = 0 but what's the E talking about? what am i trying to do...
  16. P

    Question about differentiability

    I study Calculus by myself, and I tried to solve the following question. I got the answer, but is my solution consistent? Thank you in advance. 1. The problem statement Let f be a function such that |f(x)| ≤ x² for every x. Show that f is differentiable in 0 and that f'(0) = 0. 2...
  17. B

    Caucy-Riemann equations and differentiability question

    I'm doing a little self study on complex analysis, and am having some trouble with a concept. From Wikipedia: "In mathematics, the Cauchy–Riemann differential equations in complex analysis, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential...
  18. M

    Differentiability of coefficients for 2nd order DE

    Homework Statement Argue that if y=f(x) is a solution to the DE: y'' + p(x) y' + q(x) y = g(x) on the interval (a,b), where p, q, and g are each twice-differentiable, the the fourth derivative of f(x) exists on (a,b). Homework Equations The Attempt at a Solution Its a general...
  19. G

    Proving a claim regarding differentiability

    Let F(x,y,z) be a function which is defined in the point M_0(x_0,y_0,z_0) and around it and the following conditions are satisfied: 1. F(x_0,y_0,z_0)=0 2. F has continuous partial derivatives in M_0 and around it 3. F'_z(x_0,y_0,z_0)=0 4. gradF at (x_0,y_0,z_0) != 0 5. It is known that...
  20. T

    Multivariable chain rule and differentiability

    Homework Statement Hi I'm currently trying to revise for a Calculus exam, and have very little idea of how to do the following: Let f be defined by f(x,y) = (y+e^x, sin(x+y)) Let g be of class C2 (twice differentiable with continuous second derivatives) with grad(g)(1,0) = (1,-1) and Hg(1,0)...
  21. G

    Differentiability on an Open Interval

    Hi all, I am having a little trouble understanding one of the concepts presented in my calculus class. I do not understand how the endpoints of an open interval can be differentiable. My teacher says that the endpoints of a closed interval can not be differentiable because the limit can not...
  22. D

    Proving Differentiability of a Continuous Function at x=0

    Homework Statement "A real valued function, f, has the following property: \left|f\right| is differentiable at x=0 Prove that if we specify that f is continuous at 0, then f is also differentiable at 0." Homework Equations Since \left|f \right| is differentiable we know the...
  23. T

    Proving Differentiability: Vector Calculus Homework

    Homework Statement [PLAIN]http://img261.imageshack.us/img261/1228/vectorcalc.png Homework Equations The Attempt at a Solution f({\bf a+h})-f(\bf{a})={\bf c\times h} + \|{\bf a+h} \| ^2 {\bf c} - \|{\bf a}\| ^2 {\bf c} f({\bf a+h})-f({\bf a}) = {\bf c} \times {\bf h} + {\bf...
  24. M

    Limits, Differentiability, Continuity

    Homework Statement Suppose that f is differentiable in some interval containing "a", but that f' is discontinuous at a. a.) The one-sided limits lim f'(x) as x\rightarrow a+ and lim f'(x) as x\rightarrowa- cannot both exist b.)These one-sided limits cannot both exist even in the sense of...
  25. T

    Understanding Differentiability in Vector Calculus: Homework Help and Solutions

    Homework Statement [PLAIN]http://img261.imageshack.us/img261/1228/vectorcalc.png Homework Equations The Attempt at a Solution I have the definition but what do I do with f({\bf a+h})-f(\bf{a})={\bf c\times h} + \|{\bf a+h} \| ^2 {\bf c} - \|{\bf a}\| ^2 {\bf c} ?
  26. W

    Continuity And Differentiability

    Homework Statement So I am to prove that cosine is continuous on R and differentiable on R. I already proved it for sine which was simple by using the identity of sin(x +- y)=sin(x)cos(y)+-cos(x)sin(y) Now I need to prove it for cosine and also we cannot use the identity of...
  27. U

    Function ƒ(x): Continuity & Differentiability

    Homework Statement Let f be the function defined as ƒ(x)={ lx-1l + 2, for x<1, and ax^2 + bx, for x (greater or equal to) 1, where a and b are constants. Homework Equations A) If a=2 and b=3, is f continuous for all of x? B) Describe all the values of a and b for which f is a...
  28. S

    Showing differentiability at 0

    Homework Statement Show that f(x,y) = |xy| is differentiable at 0.Homework Equations The Attempt at a Solution I thought absolute value functions are not differentiable at 0?
  29. Telemachus

    Demonstration of the differentiability of a continuous function

    Homework Statement I have some doubts about the demonstration of the differentiability. If I'm asked to proof that an average function is differentiable on all of it domain, let's suppose its a continuous function on all of its domain, but it has not continuous partial derivatives. How should I...
  30. S

    How Do We Understand Differentiability and Gradients in Multivariable Calculus?

    For a function ƒ defined on an open set U having the point X:(x1,x2,...,xn) and the point ||H|| such that the point X + H lies in the set we try to define the meaning of the derivative. \frac{f(X \ + \ H) \ - \ f(X)}{H} is an undefined quantity, what does it mean to divide by a vector...
  31. R

    How can we prove f(x,y) is differentiable using induction?

    Consider the real function f(x,y)=xy(x2+y2)-N,in the respective cases N = 2,1, and 1/2. Show that in each case the function is differentiable (C\omega) with respect to x, for any fixed y-value. whats the strategy for proving C\omega-differentiability here? i have to show with induction that f...
  32. W

    Differentiability of f(x+y)=f(x)f(y)

    Given: f(x+y)=f(x)f(y). f'(0) exists. Show that f is differentiable on R. At first, I tried to somehow apply the Mean Value Theorem where f(b)-f(a)=f'(c)(b-a). I ended up lost... Then I tried showing f(0)=1, because f(x-0)=f(x)f(0) and f(x) isn't equal to 0. However, with that...
  33. M

    Condition for differentiability

    What's the condition for f(x,y) to be differentiable in its domain? I googled for it but couln't find... Thanks in advance.
  34. H

    Differentiability and Continuity

    Hi, I don't understand why mathematicians would need to define the mathematical concepts of diffferentiabilty and conitnuity. To be honest, I don't even understand why "f(x) tends to f(a) as x tends to a" describes continuity. Also, I am wondering why f(x) = mod x is not differentiable at...
  35. J

    Differentiability of a complex function.

    (PROBLEM SOLVED) I am trying to think of a complex function that is nowhere differentiable except at the origin and on the circle of radius 1, centered at the origin. I have tried using the Cauchy-Riemann equations (where f(x+iy)=u(x,y)+iv(x,y)) \frac{\partial u}{\partial x}=\frac{\partial...
  36. L

    Is the Function x->|x| Differentiable at 0?

    I want to show x->abs(x) is not differentiable at 0 Some techniques in analysis are required... how should i do?
  37. E

    Proving Differentiability and Continuity of f'(x)

    Homework Statement show f(x)=\left\{e^{\frac{-1}{x}} \\\ x>0 f(x)=\left\{0 \\\ x\leq 0 is differentiable everywhere, and show its derivative is continuous Homework Equations Product Rule and Chain Rule for derivatives. Definition of a derivative f^{'}(a)=\frac{f(x)-f(a)}{x-a}...
  38. R

    Proving Interval Inclusion & Differentiability of f: I→R

    Let I be an open interval in R and let f : I → R be a differentiable function. Let g : T → R be the function defined by g(x, y) =(f (x)−f (y))/(x-y) 1.Prove that g(T ) ⊂ f (I) ⊂ g(T ) (The last one should be the closure of g(T), but I can't type it here) 2. Show that f ′ (I) is an interval...
  39. M

    Differentiability + Continuity?

    Homework Statement Suppose a>0 is some constant and f:R->R is given by f(x) = |x|^a x sin(1/x) if x is not 0 f(x) = 0 if x=0 for which values of a is f differentiable at x=0? Use calculus to determine f'(x) for x is not equal to 0. For what values of a is f' a continuous function defined...
  40. S

    Check differentiability in function

    Homework Statement Hello, I can't find any way to prove if this funtion is or isn't differentiable in if (x,y)=(0,0) : {f(x,y)=\displaystyle\frac{x^{3}}{x^{2}+y^{2}}} if (x,y) \neq(0,0) f(x,y)=0 if (x,y)=(0,0) The Attempt at a Solution Partial derivatives don't exist...
  41. N

    Differentiability on a closed interval

    Homework Statement Hi all I wish to show differentiability of g(x)=x on the interval [-pi, pi]. This is what I have done: g'(a) = \mathop {\lim }\limits_{h \to 0} \frac{{g\left( {a + h} \right) - g\left( {a} \right)}}{h} \\ = \mathop {\lim }\limits_{h \to 0} \frac{h}{h} \\ = 1...
  42. N

    Absolute value and differentiability

    Homework Statement Hi all I have f(x)=|x|. This I write as f(x) = -x for x<0 f(x) = x for x>0 f(x) = 0 for x=0 If I want to show that f(x) is not differentiable at x=0, then is it enough to show that f'(x) = -1 for x<0 f'(x) = 1 for x>0 and from this conclude that it is...
  43. M

    Continuity of partial derivatives in a ball implies differentiability

    Hi all, I'm looking at the following problem: Suppose that f:\mathbb{R}^2\to\mathbb{R} is such that \frac{\partial{f}}{\partial{x}} is continuous in some open ball around (a,b) and \frac{\partial{f}}{\partial{y}} exists at (a,b): show f is differentiable at (a,b). Now I know that if both...
  44. N

    Simple dirichlet function differentiability

    Homework Statement f(x) = {x, x rational, 0, x irrational g(x) = {x^2, x rational, 0, x irrational Show that f(x) is not differentiable at 0. Show that g(x) is differentiable at 0 Homework Equations f'(x) = lim(h->0) f(x+h) - f(x)/h I suppose The Attempt at a Solution Just...
  45. H

    Prove differentiability and continuity

    Homework Statement Determine that, if f(x) = {xsin(1/x) if x =/= 0 {0 if x = 0 that f'(0) exists and f'(x) is continuous on the reals. (Sorry I can't type the function better, it's piecewise) Homework Equations The Attempt at a Solution For f'(0) existing, For x ≠ 0...
  46. S

    Proof by induction of polynomial differentiability

    Homework Statement Prove that (ax^n)' = nax^n-1 using induction. I am very weak with induction proof, and I haven't had much trouble proving the basis step, but I can't seem to finish it... Homework Equations The Attempt at a Solution 1. Prove (ax)' = a (a(x+h) - a(x))/h =...
  47. G

    Differentiability of xy function

    Homework Statement Dear all, How can I show that the function f(x,y)=xy is differentiable? Thanks Dimitris Homework Equations The Attempt at a Solution
  48. A

    Sufficient condition for differentiability of a function of two variables

    Is there a convenient sufficient condition for knowing whether a function of two variables is differentiable? Isn't it something like if both the partial derivatives exist and are continuous, you know the derivative \mathbf{D}f exists?
  49. O

    Understanding Rolle's Theorem: Continuity & Differentiability

    Hallo. If we consider Rolle's Theorem: "If f is continuous on [a, b], differentiable in (a,b), and f (a) = f (b), then there exists a point c in (a, b) where f'(c) = 0." Why do we need to state continuity of f in interval and differentiability of f in open segment? Why can't we say f...
  50. P

    Condition for differentiability of a function

    given a function F:S-->R such that for every element belonging to "S" has both left hand derivative and right hand derivative and are equal to the derivative at that point. Can we say that the function is differentiable..?
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