A function is not differentiable at any point where it lacks a single, well-defined slope. This happens in four classic situations: at a discontinuity, at a sharp corner or kink, at a cusp, or at a vertical tangent. Each one breaks the derivative in a slightly different way, but they all share the same root cause: the limit that defines the derivative either doesn’t exist or isn’t finite.
Why Continuity Comes First
Before you can even ask whether a function is differentiable at a point, the function has to be continuous there. This isn’t just a guideline. It’s a theorem: if a function is differentiable at a point x = c, then it is continuous at x = c. The contrapositive is what matters for spotting non-differentiability. If a function is not continuous at a point, it cannot be differentiable there. Period.
So any type of discontinuity kills differentiability immediately. A jump discontinuity (like a step in a staircase function), a removable discontinuity (a hole in the graph), or an infinite discontinuity (where the function blows up to infinity) all make the derivative impossible at that point. The slope calculation requires you to examine how the function’s output changes as you nudge the input by a tiny amount, and if the function isn’t even approaching a single value at that point, there’s no meaningful slope to compute.
The important flip side: continuity does not guarantee differentiability. A function can be perfectly continuous at a point and still fail to have a derivative there. The remaining three cases all involve continuous functions.
Corners and Kinks
The most familiar example is f(x) = |x| at x = 0. The graph makes a sharp V shape at the origin, and that sharp turn is exactly where the derivative breaks down.
Here’s why. The derivative at a point is defined as a two-sided limit. You approach from the left and from the right, and both sides need to agree. For |x| at x = 0, if you approach from the right (using small positive values of x), the function equals x, and the slope is +1. If you approach from the left (using small negative values), the function equals -x, and the slope is -1. Since +1 and -1 aren’t equal, the two-sided limit doesn’t exist, and neither does the derivative.
This generalizes to any point where a graph abruptly changes direction. Piecewise functions are common offenders. Wherever two pieces meet at an angle rather than blending smoothly, you get a corner. The left-hand derivative and the right-hand derivative both exist, but they disagree. That mismatch is the signature of a corner.
Cusps
A cusp looks similar to a corner on a graph, but the behavior is more extreme. Instead of the slope approaching two different finite values from each side, the slope approaches infinity from one side and negative infinity from the other (or both approach the same infinity).
The standard example is f(x) = x2/3 at x = 0. The derivative of this function is (2/3)x-1/3. As x approaches 0 from the right, that derivative shoots to positive infinity. As x approaches 0 from the left, it plunges to negative infinity. The graph comes to an infinitely sharp point, like the tip of a thorn. Because the slope becomes unbounded rather than settling on a finite value, the derivative doesn’t exist.
The visual difference between a corner and a cusp: at a corner, the graph changes direction at a definite angle (like the tip of a V). At a cusp, both sides of the graph become nearly vertical as they approach the point, creating a much sharper, pointed shape.
Vertical Tangents
A vertical tangent occurs when the graph is smooth and doesn’t change direction, but the slope still becomes infinite. The function f(x) = x1/3 (the cube root of x) at x = 0 is the go-to example. Its derivative is (1/3)x-2/3, which approaches positive infinity from both sides as x goes to 0. Unlike a cusp, there’s no sign change. The graph passes smoothly through the point, but the tangent line at that point is perfectly vertical.
A vertical line has an undefined slope, so the derivative doesn’t exist. The function is still continuous at x = 0, and the graph doesn’t have a sharp turn. It’s just too steep. If you think of the derivative as a rate of change, a vertical tangent means the output is changing infinitely fast relative to the input at that instant, which the derivative can’t express as a finite number.
At the Edges of the Domain
A function can only be differentiable where it’s defined, and the standard derivative requires approaching a point from both sides. At an endpoint of a closed interval, you can only approach from one direction. For example, if a function is defined on [0, 5], the usual two-sided derivative doesn’t exist at x = 0 or x = 5.
Some textbooks and courses define one-sided derivatives at endpoints, which can be useful. But under the standard definition that most calculus courses use, a function on a closed interval is only differentiable on the open interior. This is why many theorems in calculus (like the Mean Value Theorem) require a function to be continuous on [a, b] but only differentiable on (a, b).
How to Spot Non-Differentiable Points
If you’re working with a graph, look for any place where the curve has a sharp point, a vertical tangent, or a break. These are your non-differentiable points. If you’re working with a formula, the process is more systematic:
- Check continuity first. Find any points where the function is undefined, has a hole, or jumps. The function isn’t differentiable at any of them.
- Check piecewise boundaries. Wherever two pieces of a piecewise function meet, compute the derivative from the left and the derivative from the right. If they don’t match, you have a corner.
- Check where the derivative formula breaks down. If the derivative expression involves division by zero or produces an infinite limit at some point, you likely have a cusp or vertical tangent. Look at the sign of the slope from each side to tell them apart: opposite signs mean a cusp, same sign means a vertical tangent.
Functions With No Differentiable Points at All
Most functions you encounter in a calculus course are differentiable almost everywhere, failing only at isolated trouble spots. But mathematicians have constructed functions that are continuous at every single point yet differentiable at none of them. The most famous is the Weierstrass function, built by adding up infinitely many cosine waves at increasingly high frequencies. The result is a curve that is unbroken (continuous everywhere) but so jagged at every scale that no point has a well-defined slope.
This isn’t just a mathematical curiosity. Brownian motion, the random jittering path of a particle suspended in fluid, traces a path that is continuous but nowhere differentiable. The particle has a position at every moment in time, so the path never “breaks,” but it changes direction so erratically that you can never assign it a definite velocity at any instant. Stock price charts, coastline shapes, and many natural phenomena share this kind of fractal roughness, where zooming in reveals just as much jaggedness as the full-scale view.