# Does bounded derivative always imply uniform continuity?

• lonewolf5999
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.
lonewolf5999
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!

This indeed looks like a valid counterexample!

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.

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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