f is continuous if
The limit as x approaches a exists for f(x), the limit as x approaches a equals f(a) and it is continuous along the interval if true for every point a in the Interval.
f is differentiable if
The limit as h as f evaluated at x plus h minus f evaluated at x over h exists. Along an interval if limit exists for every point a in the interval.
A differentiable function is always continuous, but not the other way around. A continuous function can not have a derivative at a point (Break or sharp corner)