The properties of a differentiable function

Question

Let $f:(0,1)\to \Bbb R$ be a differentiable function. Which of the following statements is true?

(a)If $f'$ is bounded, then $\lim_{x\rightarrow 0^{+}}f(x)<\infty$

(b)If $f$ is uniformly continuous, then $f'$ is bounded.

Answer 1

(a) True: Since $f'$ is bounded, if follows the Lipschitz condition, i. e. we can find an M, $|f(x)-f(x_0)| \le M|x-x_0|$. Therefore the function is uniformly continuous and therefore the function is bounded.

(b) False: A counter example is $f(x)=\sqrt x $. As $x \to 0$, $f'(x) \to -\infty $.

If you are having problems with these, you should try reviewing the Litpschitz condition. It should be the first instinct whenever you are dealing with bounded derivatives.

Answer 2

Just a few helpful (hopefully) remarks:

Fix $x_0$

Since $f'$ is bounded we know $\frac{f(x)-f(x_0)}{x-x_0}\leq M$ for some $M$.

Since $f$ is differentiable: for every $\epsilon > 0$ there is a $\delta > 0$ such that $|f(x)-f(x_0)|< \epsilon$ whenever $|x-x_0|<\delta$.