3.2 – Differentiability

A function does not have a derivative where:

1. There is a corner, where there are two different one-sided derivatives
2. There is a cusp, where the derivatives approach $\pm \infty$ from both sides (like a corner)
3. There is a vertical tangent
4. There is a **discontinuity

Continuity

Differentiability Implies Continuity

Since a the derivative of a function does not exist where there is a discontinuity, a differentiable function must be continuous. Therefore, a function $f$ that has a derivative at $x=a$ is continuous at $x=a$.

However, the opposite does not hold true. A piecewise function may be continuous but also have a corner at some point, which would not be differentiable.

Intermediate Value Theorem

The IVT applies to derivatives.

For a function $f(x)$ differentiable on the closed interval $[a,b]$, $f^{\prime }(x)$ takes on all values between $f^{\prime }(a)$ and $f^{\prime }(b)$.