Is $\cos^3{x}$ uniformly continuous?

Question

I've found proofs that a continuous mapping from a bounded set to $\mathbb{R}$ is uniformly continuous, but is it true the other way? If not, how do you go about proving or disproving uniform continuity?

Answer 1

When you have to claim if a function is UC or not you have also to say where it is. In this case we have two situations: A) Bounded sets B) Unbounded sets

Let $f(x)= cos^3(x)$ Case A) If we have a bounded set $A \subset \mathbb{R}$ then it is contained in the interval $ [-n,n]$ for some $n \in \mathbb{N}$ and now, applying Heine-Cantor theorem, we have the uniform continuity of $f(x)$

Case B) In this case we can consider $\mathbb{R}$ as the only set remained, we can check that the derivative of $f(x)$ which is $f'(x)= -3cos^2(x)sin(x)$ is bounded, and so we can see, using the mean value theorem, that the function is UC