proving that f' is bounded

The problem I am trying to solve is

$\displaystyle f:\Re\rightarrow\Re$ is a differentiable function and it has a continuous derivative. I need to prove that$\displaystyle f'$ is bounded on $\displaystyle [a,b]$ where $\displaystyle -\infty<a<b<\infty$.

I have shown that if $\displaystyle f:X\rightarrow Y$ is continous on $\displaystyle X$ and if $\displaystyle A$ is a compact subset of $\displaystyle X$ that $\displaystyle f(A)$ is also compact. But have no clue if this will help of where to go from here.

Thanks in advance

Re: proving that f' is bounded

Quote:

Originally Posted by

**klw289** The problem I am trying to solve is

$\displaystyle f:\Re\rightarrow\Re$ is a differentiable function and it has a continuous derivative. I need to prove that$\displaystyle f'$ is bounded on $\displaystyle [a,b]$ where $\displaystyle -\infty<a<b<\infty$.

I have shown that if $\displaystyle f:X\rightarrow Y$ is continous on $\displaystyle X$ and if $\displaystyle A$ is a compact subset of $\displaystyle X$ that $\displaystyle f(A)$ is also compact. But have no clue if this will help of where to go from here.

It tells you "it has a __continuous__ derivative..

Any continuous function is bounded on a compact set.

Any interval $\displaystyle [a,b]$ is compact.