Is a function strictly increasing if its derivative is positive at all point but critical points? $f: (a,b) \to \Bbb{R}$ is differentiable and $f'(x)>0$ at all points but at $c$ where $f'(c) = 0$.
I need to prove that $f$ is strictly increasing.
I thought to split the intervals to $(a,c)$ and $(c,b)$ and use the continuity of $f$ at $c$, but I'm not sure how to explain that.
More generally, I understand that this is true for a finite number of critical points, how do I explain that too?
Help please 
 A: Hint: You can show that $f(x)<f(c)<f(y)$, when $x<c<y$ using contradiction and mean value theorem. This is sufficient to prove that $f$ is strictly increasing everywhere.
For finite number of $\{c_i\}_{i\in I}$, $I = \{1,\ldots,n\}$ such that $a = c_0 < c_1<c_2<\ldots<c_n < c_{n+1} = b$ and $f'(c_i)=0,\ i\in I$, $f'(x)>0$ when $x\neq c_i,\ i\in I$, consider invervals $A_i=(c_{i-1},c_{i+1}),\ i\in I$. By the result for one point, $f$ is strictly increasing on each $A_i$ and since they cover $(a,b)$, it is strictly increasing everywhere.
A: Yes, it can be done for a finite number of dicontinuities: take $x<y$ and pick $z\in (x,y)$ such that no point in $(x,z)$ has derivative $0$.
You can prove each of the following inequalities with the mean value theorem:
$f(x)<f(z)\leq f(y)$.
A: Let $x_1<x_2\in \left( a,b\right)$.
First let's prove under the assumption that  $f^\prime > 0$ on $\left( a,b \right)$.
Hint 1

 Lagrange's mean value theorem tells us there is an $x\in \left( x_1, x_2 \right)$ such that $$f^{\prime}\left(x\right)=\frac{f\left(x_{2}\right)-f\left(x_{1}\right)}{x_{2}-x_{1}}$$

Hint 2

 $f^{\prime}\left(x\right)>0$ so we have $f\left(x_{2}\right)-f\left(x_{1}\right)>0$

So $f$ is strictly monotonic increasing on $\left( a,b \right)$.
Let's now weaken our assumption to the original one presented. Examining the cases where $x_1 < c < x_2$ and $\left(x_{1}<x_{2}\leq c\right)\vee\left(c\leq x_{1}<x_{2}\right)$ seperately sounds like a good idea, and we should be able to use the above result to prove them quite easily!
Can you see how to proceed from here?
As for a finite number of critical points: I'd try induction on the number of critical points.
