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. D

    Continuity and Differentiability

    Homework Statement Can someone tell me why and why not following functions are Continous and Differentiable. I am also providing the answer but can some help me understand.. thanks 1) f(x) = x^(2/3) -1 on [-8,8] answer: function is continuous but not differentiable on -8. Is that...
  2. W

    Differentiability in nature (how many levels typically occur? )

    Here's a question I've thought about on several occasions: How many levels of derivatives (rates of change) typically occur for objects in nature? For instance, a car has a position, velocity (1st derivative), and acceleration (2nd derivative), but it can also be said to have a rate of...
  3. C

    Analysis differentiability and limits -

    [b]1. Suppose that f is differentiable on R and \lim_{x \rightarrow \infty} f'(x) = M. Show that \lim_{x \rightarrow \infty} (f(x+1)-f(x)) also exists, and compute it. [b]3. I am pretty sure the limit will be equal to m. Here is my attempt. \lim_{x \rightarrow \infty} f'(x) = \lim_{x...
  4. ?

    Continuity, Differentiability, and \mathbb{N}: Showing an Inequality

    This isn't homework per se... It's a question from a book I'm self-studying from. If f is continuous on [a,b] and differentiable at a point c \in [a,b], show that, for some pair m,n \in \mathbb{N}, \left | \frac{f(x)-f(c)}{x-c}\right | \leq n whenever 0 \leq |x-c| \leq \frac{1}{m}...
  5. D

    Proving Differentiability of f(x,y) at (0,0)

    Ok, so I have f(x,y)=(p(x)+q(y))/(x^2+y^2) where (x,y)NOT=0 and f(0,0)=0. the basic idea of the function is that the numerator contains 2 polynomials>2nd order. and the denominator has a Xsquared+ysquared. I have to prove that if f(x,y) is differentiable at (0,0) then its partial derivatives...
  6. C

    Analysis- continuity and differentiability

    Hi, could somebody please help me with the following question, I have been stuck on it for ages. [b]1. let f[0,1] -> R be continuous with f(0)=0, f(1)=1. Prove the following: a.(i) If for c in (0,1) f is differentiable at c with f'(c)<0 then there are exists points y such that f(x)=y has...
  7. M

    Lipschitz Condition and Differentiability

    Let K>0 and a>0. The function f is said to satisfy the Lipschitz condition if |f(x)-f(y)|<= K |x-y|a .. I am given a problem where I must prove that f is differentiability if a>1. I know I need to show that limx->c(f(x)-f(c))/ (x-c) exists. I am having quite a hard time. Any hints?
  8. M

    Differentiability and convergence.

    Let f:(a,infinity)-->R (reals) be differentiable and let A and B be real numbers. Prove that if f(x)-->A and f '(x)-->B as x --> infinity, then B=0. I would guess that the mean value theorem may be needed but I am not sure how to use it considering that we're dealing with x --> infinity.
  9. T

    Is Taylor's Series the Key to Proving Differentiability for sinx/x?

    Let f(x)= sinx/x if x \neq 0 and f(0)=1 Find a polynomial pN of degree N so that |f(x)-pN(x)| \leq |x|^(N+1) for all x. Argue that f is differentiable, f' is differentiable, f" is differentiale .. (all derivatives exist at all points). I'm not sure about this one at all. Can you guys...
  10. T

    Applying Landau's Inequality to Prove Bounds on f'(x)

    a. Suppose f is twice differentiable on (0, infinity). Suppose that |f(x)|< or equal A0 for all x>0 and that the second derivative satisfies |f''(x)|< or equal A2 for all x>0. Prove that for all x>0 and all h>0 |f'(x)| < or equal 2A0/h + A2h/2 This is sometimes called Landau's inequality...
  11. M

    Help with calculus problem- differentiability, continuity, with variables

    Homework Statement y= y= {1+3ax+2x^2} if x is < or = 1 {mx+a} if x>0 what values for m and a make x continuous and differentiable at 1? Homework Equations n/a The Attempt at a Solution i solved for when x=1. i got 3+3a. this is also the right hand...
  12. S

    Finding Values of a & b for Differentiability of f(x)

    Homework Statement Find the values of the constants a and b such that the function f(x) is differentiable on R Homework Equations f(x) = ax2 if x < 2 f(x) = -4(x-3) + b if x >= 2 The Attempt at a Solution ax2 = -4(x-3) + b 2xa = -4x...
  13. N

    Proof of Theorem on Differentiability and Lebesque Integral

    Hello, I was wondering where I can find a proof to the following theorem: If F is differentiable at every pt. of [a,b] and if F' is lebesque on [a,b] then F(x) = int(f,dt,a,x) a<=x<=b. And the converse. He gives the theorem are page 324 and a reference in his bibliography. I...
  14. J

    When Does the Collection Approach Zero Uniformly for Differentiable Functions?

    Let f:]a,b[\to\mathbb{R} be a differentiable function. For each fixed x\in ]a,b[, we can define a function \epsilon_x: D_x\to\mathbb{R},\quad\quad \epsilon_x(u) = \frac{f(x+u) - f(x)}{u} \;-\; f'(x) where D_x = \{u\in\mathbb{R}\backslash\{0\}\;|\; a < x+u < b\}. Now we have...
  15. A

    What are the Cauchy-Riemann equations and their geometric interpretation?

    Is there a geometric meaning for the derivative of a complex valued function, or any other motivation for the derivative?
  16. M

    Proving Differentiability of f given g'(x) < 0 $\forall$ x

    Suppose the real valued g is defined on \mathbb{R} and g'(x) < 0 for every real x. Prove there's no differentiable f: R \rightarrow R such that f \circ f = g.
  17. N

    Differentiability implies continuity proof (delta epsilon)

    1. The problem statement. Give a complete and accurate \delta - \epsilon proof of the thereom: If f is differentiable at a, then f is continuous at a. 2. The attempt at a solution Known: \forall\epsilon>0, \exists\delta>0, \forall x, |x-a|<\delta \implies \left|\frac{f(x) - f(a)}{x-a}...
  18. E

    Proving Differentiability of f in R$^2$

    Define f(0,0) = 0 and f(x,y) = \frac {x^3}{x^2 + y^2} if (x,y) \neq (0,0) a) Prove that the partial derivatives of f are bounded functions in R^2. b) Let \mathbf{u} be any unit vector in R^2. Show that the directional derivative (D_{\mathbf{u}} f)(0,0) exists, and its absolute value is at...
  19. A

    Understanding Differentiability of Functions with Several Variables

    Hello: These are not h/w problems but something from the class notes which I am not able to fully understand. I have two questions stated below Question1 We are doing chain rule and function of several variables. To explain the prof has first explained about single variables and then...
  20. G

    Basic differentiability question

    Homework Statement Let f: R -> R be a continuous function such that f '(x) exists for all x =/= 0 . Say also that the limit of f '(x) as x goes to 0 exists and is equal to L. Must f '(0) exist as well? Prove or disprove. The Attempt at a Solution I can't come up with a proof or...
  21. E

    Complex differentiability problem

    Hi guys, I have this problem understanding that holomorpic functions must be infinitely differentiable. Indeed, it does follow from the Cauchy formula. But take z=x+iy. It satisfies C-R equations and has a first derivative = 1. I fail to see how this function is infinitely differentiable...
  22. G

    Solving 2 Problems: Finding Critical Points & Differentiability

    I have two problems. I posted the first problem before but I still can´t solve it. Homework Statement Find and classify the critical points of f(x,y,z) = xy + xz + yz + x^3 + y^3 + z^3 Homework Equations - The Attempt at a Solution df/dx = y + z +3x^2, df/dy = x + z + 3y^2...
  23. M

    Differentiability and continuity

    Hi. How do I show that f is differentiable, but f' is discontinuous at 0? I guess I'm just looking for a general idea to show discontinuity. Thanks
  24. V

    Differentiability of Max Function?

    Homework Statement Homework Equations The Attempt at a Solution I think to determine where it's differentiable it has something to do with partial derivatives. But I am just so completely clueless on how to even start this guy off that any tips or minor suggestions on where to...
  25. S

    Proving Continuity of a Function Using the Definition of Differentiability

    Homework Statement Suppose f:(0,\infty)->R and f(x)-f(y)=f(x/y) for all x,y in (0,\infty) and f(1)=0. Prove f is continuous on (0,\infty) iff f is continuous at 1. Homework Equations I think I ought to use these defn's of continuity: f continuous at a iff f(x)->f(a) as x->a or f is cont at a...
  26. L

    Why does differentiability imply continuity?

    I've been thinking... Since derivatives can be written as: f'(c)= \lim_{x\rightarrow{c}}\frac{f(x)-f(c)}{x-c} and for the limit to exist, it's one sided limits must exist also right? So if the one sided limits exist, and thus the limit as x approaches c (therefore the derivative at c)...
  27. R

    How to prove differentiability?

    Homework Statement Given an interval I. The function f goes from I to the real line. Define f as f(x)=x^2 if x rational or 0 if x is irrational. show that f i differentiablle at x=0 and find its derivative at this point, i.e. x=0. Homework Equations I have a given lead on this. That...
  28. quasar987

    Measure zero and differentiability

    [SOLVED] measure zero and differentiability Homework Statement I proved in the preceding exercise the following characterization of measure zero: "A subset E of R is of measure zero if and only if it has the following property: (***) There exists a sequence J_k=]a_k,b_k[ such that every x in...
  29. H

    Differentiability of function at only 0

    1. function is x^2 if x is rational 0 if x is irrational. I need to prove that function is only differentiable at 0. 2. f'(x)=lim(h->0)=(f(x+h)-f(x))/h 3. fruitless attempt-----> So f'(0)=lim(h->0) f(h)/h=lim(h->0)x since it's 0 when irrational and x when rational, 0 when irrational, x=0...
  30. K

    Differentiability and parametric curves

    f(t)=(t^3, |t|^3) is a parametric representation of y=f(x)=|x|. Consider y=|x|, the left hand derivative f '-(0)=-1 and the right hand derivative f '+(0)=1, so f(x) is clearly not differentiable at 0. But f '(t)=(3t^2, 3t^2) for t>=0 f '(t)=(3t^2, -3t^2) for t<=0 f '(0)=(0,0) and f(t)...
  31. S

    CR equations and differentiability

    Homework Statement Where is f(z) differentiable? Analytic? f(z) = x^{2} + i y^{2} Homework Equations Cauchy-Riemann Equations The Attempt at a Solution I calculated the partial derivatives, u_{x} = 2x v_{y} = 2y u_{y} = 0 v_{x} = 0 Then said that for the CR equations to...
  32. J

    Differentiability of the mean value

    So if a function f:[a,b]\to\mathbb{R} is differentiable, then then for each x\in [a,b] there exists \xi_x \in [a,x] so that f'(\xi_x) = \frac{f(x)-f(a)}{x-a} Sometimes there may be several possible choices for \xi_x. My question is, that if the mapping x\mapsto \xi_x is...
  33. P

    Differentiability and continuity confusion

    Hi there just a general question: this involves continuity and differentiability suppose: f(x) = sin 1/x if x not equal to 0 f(0) = 0 PROVE F IS NOT DIFFERENTIABLE AT 0 i understand if it is not differentiable at 0 then it may not be continuous at 0. however is there...
  34. C

    Analyzing Continuity and Differentiability of f(x) at x=1 & x=3

    Homework Statement f(x) is a piecewise function defined as: |x-3| x>=1 \frac{x^2}{4}-\frac{3x}{2}+\frac{13}{4} x<1 Discuss the continuity and differentiability of this funtion at x=1 and x=3 Homework Equations The Attempt at a Solution At x=3, this function is continuous...
  35. K

    Continuity and Differentiability of g Defined by Integrals

    Theorem: Let f be continuous on [a,b]. The function g defined on [a,b] by http://tutorial.math.lamar.edu/AllBrowsers/2413/DefnofDefiniteIntegral_files/eq0051M.gif is continuous on [a,b], differentiable on (a,b), and has derivative g'(x)=f(x) for all x in (a,b) 1) Given that g is defined...
  36. quasar987

    Definition of differentiability on a manifold

    My text defines differentiability of f:M\rightarrow \mathbb{R} at a point p on a manifold M as the differentiability of f\circ \phi^{-1}:\phi(V) \rightarrow \mathbb{R} on the whole of phi(V) for any chart (U,\phi ) containing p, where V is an open neighbourhood of p contained in U. Is this...
  37. D

    Proving differentiability of function on a Lie group.

    On page 116 of Choquet-Bruhat, Analysis, Manifolds, and Physics, Lie groups are defined, and the first exercise after that asks you to prove that for a Lie group G f:G \rightarrow G; x \mapsto x^{-1} is differentiable. I know from the previous definitions that a function f on a manifold...
  38. K

    Calculating Power Series Coefficients with Differentiability

    Let be the powe series: f(x)=\sum_{n=0}^{\infty}a(n)x^{n} then if f(x) is infinitely many times differentiable then for every n we have: n!a(n)=D^{n}f(0) (1) of course we don't know if the series above is of the Taylor type, but (1) works nice to get a(n) at least for finite n.
  39. P

    Showing differentiability in R^n

    When you are given a function and asked for differentiability at a point in domain or the whole domain, what's the normal procedure? I think the definition is almost useless in here... Also, I'm still stuck on the previous questions so if anyone bothers to check them out...
  40. I

    (Complex Variables) Differentiability of Arg z

    I am proving that the function f(z) = Arg z is nowhere differentiable by using the definiton of a derivative. I let z = x + yi. Then, if the limit exists, we have f'(z) = lim (/\z -> 0) ( f(z + /\z) - f(z) ) / /\z. (Note that /\ is the triangle symbol) Also, let /\z = p + iq, where p and...
  41. quasar987

    Differentiability VS derivability

    differentiability VS "derivability" In french, the quality of a function which in english is called 'differentiable', we call 'dérivable'. And we call 'différentiable' at the point (x,y) a function f such that we can write f(x+h,y+k) - f(x,y) = h*df(x,y)/dx + k*df(x,y)/dy + o(sqrt{h²+k²})...
  42. F

    Differentiability of Functions with Two Real Variables

    Am mainly stuck on parts c) and d) but thought i'd put in the other questions as an aid 2. a) Define what it means to describe a function f of two real variables as differentiable at (a, b)? Define (as limits) the partial derivatives df/dx and df/dy at (a, b) and prove that if f is...
  43. A

    Differentiability of weird function

    Let f(x) = x if x rational and f(x) = 0 if x is irrational Let g(x) = x^2 if x rational and g(x) = 0 if x is irrational. Both functions are continuous at 0 and discontinuous at each x != 0. How do I show that f is not differentiable at 0? How should I show that g is differentiable...
  44. Z

    Is Function f(x) Differentiable at x = 4?

    hey guys, what's up. Its my first time posting so yeah... Ok I'm attempting to do the following problem... let f(x) = x^3 for x is less than or eqaul to 4. (6x^2)-8x when x is greater than 4 i need to see if f(x) is differentiable at x = 4. I tried it through the...
  45. A

    Analysis Problem (proving differentiability at a point)

    Need some hints on how to go about doing this: f(x, y)=\left\{\begin{array}{cc}\frac{x^4 + y^4}{x^2 + y^2},&\mbox{ if } (x, y)\neq (0,0)\\0, & \mbox{ if } (x, y) = 0\end{array}\right. Show that f is differentiabile at (0, 0). I've tried a number of things, too ugly and not worth...
  46. himanshu121

    Continuity And Differentiability

    Consider f(x)=x^3-x^2+x+1 g(x)=\left\{\begin{array}{cc}{max\{f(t),0\leq t \leq x\}}\;\ 0\leq x \leq 1 \\ 3-x\;\ 1< x \leq 2\end{array}\right Discuss the continuity and differentiability of g(x) in the interval (0,2) I know how to do it As f(x) is increasing function therefore max...
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