# proving that f' is bounded

• May 5th 2012, 01:54 PM
klw289
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.

• May 5th 2012, 02:18 PM
Plato
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.