How to prove that a function is continuous? Could you give me some hint how to solve this question:
Suppose $f$ is a differentiable function for all $0<x<1$,$f(0)=1,f'(x)>0$ in the given interval.
It is obvious that $f$ is continuous for all $0<x<1$, but is it continuous at $x=0$ ?
Thanks.

First, knowing the definition of a continuous function helps.
Quoting wikipedia here,
A function f(x) is continuous at point c if the limit of f(x) as x approaches c is f(c).
You have some facts stated earlier:


*

*f(x) is differentiable in 0 < x < 1

*f(0) = 1

*f'(x)>0 in (0,1)


We should check the limit of f(x) as x approaches 0. The answer is going to be a number, whose derivative is >0. However, Since f'(0) = 0, we can automatically assume that 
lim as f(x) approaches 0 isn't equal to f(0) because their derivatives are different. 
Hence f(x) is not continuous at 0. 
Hope this makes sense.
