Does bounded derivative always imply uniform continuity?

In summary, the conversation discusses the problem of proving uniform continuity on a differentiable function with a bounded derivative on an open subset of the real numbers. A possible counterexample is presented and the importance of intervals for continuity is mentioned.
  • #1
lonewolf5999
35
0
I'm working on a problem for my analysis class. Here it is:

Let f be differentiable on an open subset S of R. Suppose there exists M > 0 such that for all x in S, |f'(x)| ≤ M, i.e. the derivative is bounded. Show that f is uniformly continuous on S.

I'm not too sure that this question is correct though, as I think I have a counterexample. Let S be the union of (-1,0) and (0, 1), then clearly S is open. Define f(x) = |x| / x for x in S.
Then if x < 0, f(x) = -1, and if x > 0, f(x) = 1.
f'(x) = 0 at every x in S, since 0 is not in S, so f is differentiable on S and the derivative is bounded.

And now f is not uniformly continuous on S, since if we set ε= 1, let δ be arbitrary, and pick x,y close to 0 such that x<0, y>0, and |x - y| < δ, it does not follow that |f(x) - f(y)| < ε. So no δ will work for this ε.

I'd really appreciate any feedback on my reasoning. Thanks for your time!
 
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  • #2
This indeed looks like a valid counterexample!
 
  • #3
indeed, the problem should read:

Let f be differentiable on an open interval S of R. Suppose there exists M > 0 such that for all x in S, |f'(x)| ≤ M, i.e. the derivative is bounded. Show that f is uniformly continuous on S.

so there's something special about intervals and continuity. intervals are connected.
 
  • #4
Thanks for the replies!
 

1. What is a bounded derivative?

A bounded derivative is a function whose derivative is limited or restricted by a specific value or range. This means that the rate of change of the function is not allowed to exceed a certain value, either positive or negative.

2. How does a bounded derivative affect the continuity of a function?

A bounded derivative does not necessarily guarantee continuity of a function. While a bounded derivative indicates that the rate of change of the function is limited, it does not necessarily mean that the function will be continuous. However, it is a necessary condition for a function to be uniformly continuous.

3. Can a function have a bounded derivative but not be uniformly continuous?

Yes, a function can have a bounded derivative but still not be uniformly continuous. This is because boundedness of the derivative is only a necessary condition, but not a sufficient one, for uniform continuity. There are other factors, such as the behavior of the function at the endpoints of the interval, that also need to be considered.

4. Are there any exceptions to the statement "bounded derivative implies uniform continuity"?

Yes, there are some exceptions to this statement. For example, if a function is defined on an open interval, then it may have a bounded derivative but not be uniformly continuous. This is because the function may have a singularity or discontinuity at one or both of the endpoints of the interval.

5. How can I determine if a function has a bounded derivative?

To determine if a function has a bounded derivative, you can take the derivative of the function and then evaluate its absolute value. If the absolute value of the derivative is less than or equal to a constant value for all points in the domain, then the function has a bounded derivative. Alternatively, you can also use the Mean Value Theorem to check for boundedness of the derivative.

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