Show that f Uniform Differentiable implies f' Uniform Continuous

In summary, uniform differentiability implies that there is a derivative of f everywhere, and that f' is continuous on (a,b).
  • #1
Aryth
8
0

Homework Statement


A function [itex]f:(a,b)\to R[/itex] is said to be uniformly differentiable iff [itex]f[/itex] is differentiable on [itex](a,b)[/itex] and for each [itex]\epsilon > 0[/itex], there is a [itex]\delta > 0[/itex] such that [itex]0 < |x - y| < \delta[/itex] and [itex]x,y \in (a,b)[/itex] imply that [itex]\left|\frac{f(x) - f(y)}{x - y}-f'(x)\right| < \epsilon[/itex].

Prove that if f is uniformly differentiable on [itex](a,b)[/itex], then [itex]f'[/itex] is continuous on [itex](a,b)[/itex].


The Attempt at a Solution



This is my first time being presented with the definition of uniform differentiability. I suppose that I am looking to show that the definition of uniform differentiability implies [itex]|f'(y) - f'(x)|< \epsilon[/itex]... However, I'm having a hard time doing that. Any help would be appreciated.
 
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  • #2
Here's one thing to think about: What is it about uniform differentiability that would give you the continuity in f'? What does it mean to be continuous in f, but discontinuous in f'? A good example would be a function like [itex]f(x) = x^2\sin(1/x), x \neq 0, f(x) = 0, x = 0[/itex]. This function is continuous everywhere but does not have a continuous derivative. Thus it is not uniformly differentiable. Why is that?
 
  • #3
tjackson3 said:
Here's one thing to think about: What is it about uniform differentiability that would give you the continuity in f'? What does it mean to be continuous in f, but discontinuous in f'? A good example would be a function like [itex]f(x) = x^2\sin(1/x), x \neq 0, f(x) = 0, x = 0[/itex]. This function is continuous everywhere but does not have a continuous derivative. Thus it is not uniformly differentiable. Why is that?

All that I have been given at this point is that if f is differentiabile at a point, then f is continuous at that point. So all I could say that it would make f uniformly continuous...

So, I guess if f is uniformly differentiable, then it has a derivative everywhere, and that should make f' be continuous everywhere.

Your explanation makes sense, but I guess I should be specific. I'm having trouble understanding the meaning of: [itex]\left|\frac{f(x) - f(y)}{x-y} - f'(x)\right| < \epsilon[/itex]
 
  • #4
I'm a bit confused...are we trying to show that f' is continuous or uniformly continous?
 
  • #5
murmillo said:
I'm a bit confused...are we trying to show that f' is continuous or uniformly continous?

Just continuous I believe... I may have misstated the problem in the title. Apologies.
 
  • #6
I don't think I've ever heard of uniformly differentiable before... but f(x)-f(y)/(x-y) is the slope of the line connecting f(x) and f(y). The inequality is saying that that slope gets arbitrarily close to the derivative of f at x. You might want to ask: how is that different from the definition of differentiable?

Here's a hint for the proof:
|f(y) - f(x)| = |f(y) - [f(y)-f(x)/(y-x)] + [f(x)-f(y)/(x-y)] - f(x)|
 
  • #7
In the first slope you put (y-x) and in the next one you put (x-y)... Is that right?
 
  • #8
Yes, but I'm actually adding and subtracting the same thing because I multiplied by -1 on the top and bottom.
 
  • #9
Ok then, I see what you're doing. I did not notice that the f(x) and the f(y) were also switched.

That helped me finish it, thank you very much.
 

1. What is the definition of uniform differentiability?

Uniform differentiability is the property of a function where the rate of change of the function can be consistently and continuously calculated at all points in its domain using a single derivative function.

2. How is uniform differentiability different from ordinary differentiability?

Uniform differentiability is a stronger condition than ordinary differentiability. While ordinary differentiability only requires the existence of a derivative at each point, uniform differentiability requires that the derivative function is also continuous.

3. What does it mean for a function to be uniformly continuous?

A function is uniformly continuous if for any small change in the input, there is a corresponding small change in the output. In other words, the function's values do not vary significantly even when the input values are close together.

4. How does uniform differentiability imply uniform continuity?

If a function is uniformly differentiable, it means that its derivative function is continuous. This means that small changes in the input will result in small changes in the output, making the function uniformly continuous.

5. What are the practical applications of uniform differentiability?

Uniform differentiability is an important concept in analysis and is used in many mathematical models, particularly in physics and engineering. It also has applications in optimization problems and numerical analysis.

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