Question: if a function is continuous then it is differentiable at that point

Answer: Yes, If f is differentiable at a point x, then f must also be continuous at x. In particular, any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly.