Example about monotonic function

Question

I am having trouble with finding examples for the follwing;

1) A function f:(-1,1) to R (real numbers) which is continuous and monotonic increasing, but not differentiable at 0. I have been thinking about taking - abs(x) but I did not see that this function would guarantee it is monotonic increasing, so it did not work.

2)A function f:R TO R which is strictly monotonic increasing and differentiable on R, with the property that it is derivative at 0 is zero. Here I used f(x)=x^3, I think it works with this one.

I really need any[ hint for part 1.

Thank you

Answer 1

For 1/ you can use a simple piece wise function:

$$ f(x)=\left\{ \begin{array}{3} x &\text{ if }& x<0 \\ 2x &\text{ otherwise}\end{array} \right. $$

Your 2/ is ok.

Answer 2

Your example in 2) is fine.

1): let $f(x)=0 $ if $x \in(-1,0)$ and $f(x)=x$ if $x \in [0,1)$