how do I show the following assertions:
- Let $f\colon I\to \mathbb{R}$ differentiable, let $a_n,b_n$ be sequences of real numbers such that $a_n \leq c \leq b_n$, then f'(c)= \lim \frac{f(b_n)-f(a_n)}{b_n - a_n}
- Let $f\colon[a,b]\to \mathbb{R}$ differentiable in $(a,b)$ with f' bounded,if $f$ has the intermediate value property then $f$ is continuous in $[a,b]$.
- Let $f(x)=\sin\left(\frac 1x\right) ,x\neq 0$ and $f(0)=0$, then $f$ has the intermediate value property.
thanks.