help with differentiable and continuous functions not just a point

Question

Can anyone help proving that a function f:(a,b)->R that is differentiable on (a,b) not just a point, is also continuous on (a,b).

I was looking into taking an arbitrary point in (a,b) and using left and right limits somehow.

Answer 1

Let $x_{0}\in (a,b)$. Suppose $f$ is differentiable at $x_{0}$. So the limit $$\lim_{x\rightarrow x_{0}}\frac{f(x)-f(x_{0})}{x-x_{0}}=f'(x_{0}) $$ exists. Obviously the limit $\lim_{x\rightarrow x_{0}}(x-x_{0})$ exists and its equal 0. By the product rule for limits, $lim_{x\rightarrow x_{0}}(f(x)-f(x_{0}))$ exists. So, we have $$\lim_{x\rightarrow x_{0}} (f(x)-f(x_{0}))=\lim_{x\rightarrow x_{0}}(x-x_{0})f'(x_{0})=0, $$ so $\lim_{x\rightarrow x_{0}}(f(x)-f(x_{0}))=0$, which proves that $f$ is continous at $x_{0}$.

This is valid for all $x_{0}\in (a,b)$, so, $f$ is continous in $(a,b)$.