# Example about monotonic function

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

• Hint: $f(x)=x$ and $f(x)=3x$ are both continuous and monotonic increasing Commented Nov 28, 2017 at 7:59
• You tried $$f(x)=\lvert x\rvert=\begin{cases}ax&\text{if }x\ge 0\\ bx&\text{if }x<0\end{cases}$$ with $a=1,\ b=-1$ and it didn't work. Try with another $b$.
– user228113
Commented Nov 28, 2017 at 7:59

$$f(x)=\left\{ \begin{array}{3} x &\text{ if }& x<0 \\ 2x &\text{ otherwise}\end{array} \right.$$
1): let $f(x)=0$ if $x \in(-1,0)$ and $f(x)=x$ if $x \in [0,1)$