Relationship+between+differentiability+and+continuity

Differentiability is the ability to obtain the slope of a tangent line at a point by utilizing the derivative of a curve.

Continuity is the instance of the limit from the left being equal to the limit from the right at an evaluated point equal to the limit. In general, continuity is absent when a derivative cannot be taken, but corners and cusps are exeptions. At these, the curve is continuous but not differentiable because the derivitive from the left does not equal the derivative from the right.