Continuity and derivatives Why does differentiability imply continuity?
For instance, consider $f(x)=x$ on $(-\infty,0]$ and $f(x)=x+1$ on $(0,\infty)$. 
Then the at 0 the left derivative equals the right derivative equals 1, so why doesn't $f'(0)=1$?

The right derivative at the origin is not $1$; it's undefined, since the line connecting the point $(h,f(h))=(h,h+1)$ (for $h>0$) to the point $(0,f(0))=(0,0)$ becomes infinitely steep as $h$ approaches zero.
