Concept preview

What you will be able to do

1. 01State and justify the theorem that if $f$ is differentiable at $x=a$ then $f$ is continuous at $x=a$, and use its contrapositive: a discontinuity forces non-differentiability.
2. 02Explain why the converse FAILS — a function can be continuous at a point yet not differentiable there — and identify the three classic culprits: corners, cusps, and vertical tangents.
3. 03Locate where a derivative fails to exist for graphs and piecewise functions by testing continuity first and then comparing left-hand and right-hand slopes.

Here is the theorem precisely: if $f$ is differentiable at $x=a$ — meaning the limit $f'(a)=\lim_{h\to 0}\dfrac{f(a+h)-f(a)}{h}$ exists as a finite number — then $f$ is continuous at $x=a$. In plain terms, a function that has a tangent line at a point cannot have a hole or a jump there. Smoothness is a stronger demand than unbrokenness, and meeting the stronger demand automatically satisfies the weaker one. This gives you a powerful diagnostic: continuity is a NECESSARY condition for differentiability, so if a function is not continuous at a point, it cannot possibly be differentiable there.