When a function is "not differentiable".

Question

A function is "not differentiable" when the graph has sharp points,cusps, and discontinuities in it

It also means that we can draw infinitely many tangents at that particular point. so there is no unique tangent at that point.

am i right? and are there any other reasons?

Answer 1

Hi Rahul,

I was going to record a video response for you, but it seems the system is not working. Your intuition is basically correct. Remember what the derivative signifies: the rate of change of the function. So, if the tangent line at a point (whose slope represents the rate of change at that point) is not defined - either because there is a discontinuity in the function or because there is a cusp etc. - then there is no defined derivative at that point and the function is not differentiable at that point.

More rigorously, a function is differentiable on the interior of a closed interval if and only if the limit lim_h->0 [f(x+h) - f(x)]/h exists at any given point x in the interval. More specifically, the left and right limits must exist AND be equal at the chosen point. This explains why discontinuities and sharp points imply that the function is not differentiable at those points. If you take the left and right hand limits, they exist but they are different - hence the limit does not exist and it is not differentiable.

I'm happy to discuss the intuition behind these concepts more. Just message me to schedule a session!

Drew