Does a Function Need to Be Continuous to Be Differentiable?

We see that if a function is differentiable at a point, then it must be continuous at that point. ... If is not continuous at , then is not differentiable at . Thus from the theorem above, we see that all differentiable functions on are continuous on .

Should a function be continuous to be differentiable?

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.

Why should a function be continuous and differentiable?

Differentiability always implies continuity because you can prove it. ... If the interval is open, since the definition of the slope of the tangent line depends on the value of the function then it needs to be continuous almost everywhere in the interval.