Non-Differentiable Points

What are non differentiable points

In the entry on derivatives, we saw that if a function $f ( x )$ is differentiable at a point $c$, then the function is continuous at that point. However, there are cases where a function is continuous at $c$ but not differentiable. More generally, the non-differentiable points of a function $f ( x )$ occur when:

• The right-hand and left-hand limits of the difference quotient exist and are finite but are not equal. \(f_{-}^{'} ( c ) \neq f_{+}^{'} ( c )\)
• The limit of the difference quotient is infinite.

These points are categorized into three main types, which we will discuss below.

Inflection point with vertical tangent

An inflection point is a point where the concavity of a function changes. In this case, we have a point of non-differentiability $c$ of the function, which results in an inflection point with a tangent parallel to the $y$-axis (a vertical tangent). At such a point, the following occurs:

This behavior indicates that the slope of the tangent becomes vertical at $x = c$ while the function may change concavity around this point. In the case shown in the figure, we have \(f_{-}^{'} ( c ) = f_{+}^{'} ( c ) = + \infty\)

If the curve were reflected across the y-axis, we would have \(f_{-}^{'} ( c ) = f_{+}^{'} ( c ) = - \infty\)

Cusps

In the case of cusps, the right-hand and left-hand limits are infinite and have opposite signs.

In the case shown in the figure, we have: \(f_{-}^{'} ( c ) = - \infty \text{and} f_{+}^{'} ( c ) = + \infty\)

If the cusp were facing upwards instead of downwards, we would have: \(f_{-}^{'} ( c ) = + \infty \text{and} f_{+}^{'} ( c ) = - \infty\)

Corners

A corner occurs when the left-hand derivative and the right-hand derivative exist but are not equal. In the case of corners points, there are two tangents to the graph at the same point, and they are different from each other.

In this case we have:

How can we verify the differentiability of a function without relying on the limit of its difference quotient?

In general, let $f ( x )$ be a function continuous on an interval ([a,b]) and differentiable on that interval, except possibly at the point $x_{0} \in [ a , b ]$. If the limits $\underset{x \rightarrow x_{0}^{-}}{lim} f^{‘} ( x )$ and $\underset{x \rightarrow x_{0}^{+}}{lim} f^{‘} ( x )$ exist, then:

if $\underset{x \rightarrow x_{0}^{-}}{lim} f ′ ( x ) = \underset{x \rightarrow x_{0}^{+}}{lim} f ′ ( x ) = ℓ$, with $ℓ \in \mathbb{R}$ then the function is differentiable at $x_{0}$, and it follows that $f^{‘} ( x_{0} ) = ℓ$.