A function is differentiable at a point when the slope of the curve at that point can be defined as a single, finite number. More precisely, the limit that defines the derivative must exist. If the graph has a sharp corner, a cusp, a vertical tangent, or a discontinuity at a point, the function fails to be differentiable there.
The Limit That Defines the Derivative
Differentiability comes down to one question: does this limit produce a real number?
f'(c) = lim as h→0 of [f(c + h) − f(c)] / h
This formula measures the slope of a tiny secant line between two points on the curve, then shrinks the gap between those points to zero. If the result settles on a single finite value, the function is differentiable at c, and that value is the derivative. You can write the same idea a different way: f'(c) = lim as x→c of [f(x) − f(c)] / (x − c). Both forms are equivalent.
The critical detail is that the limit must exist from both sides. The slope as you approach from the left (called the left-hand derivative) and the slope as you approach from the right (the right-hand derivative) must both exist and be equal. If they disagree, or if either one fails to exist, the function is not differentiable at that point.
Continuity Is Necessary but Not Enough
Every differentiable function is continuous. The contrapositive is useful: if a function isn’t continuous at a point, it’s automatically not differentiable there. A jump in the graph means the limit in the derivative formula can’t settle on a single value.
But continuity alone doesn’t guarantee differentiability. Plenty of functions pass through a point without any break in the graph yet still fail the derivative test. The absolute value function, f(x) = |x|, is the classic example. It’s perfectly continuous at x = 0, but the slope coming from the left is −1 and the slope coming from the right is +1. Those don’t match, so the derivative doesn’t exist there. Continuity gets you in the door; smoothness is what actually matters.
Four Ways Differentiability Fails
When a function is not differentiable at a point despite being defined there, the cause falls into one of a few geometric categories.
- Corners. The graph makes a sharp turn so the left-hand and right-hand slopes are different finite numbers. The absolute value function at x = 0 is the standard example. More generally, a corner is any point where a continuous function’s derivative jumps abruptly from one value to another.
- Cusps. Two branches of the curve meet at a point, and the slopes from each side head toward infinity in opposite directions (one goes to +∞, the other to −∞). The function f(x) = x^(2/3) at x = 0 behaves this way. The curve gets infinitely steep from both sides but in opposite vertical directions.
- Vertical tangents. The curve is smooth, but the tangent line at the point is perfectly vertical. A vertical line has an undefined slope because the calculation involves dividing by zero. The function f(x) = x^(1/3) at x = 0 is a typical case. The slope approaches infinity from both sides in the same direction, so there’s no finite derivative.
- Discontinuities. If the function jumps, has a hole, or goes to infinity at the point, the derivative formula breaks down entirely. You can’t measure the slope of a curve at a point where the curve isn’t connected.
Some functions also oscillate so rapidly near a point that the slope never settles. The function f(x) = x · sin(1/x) at x = 0 is a well-known example: the wiggles compress infinitely as you approach zero, and the derivative limit fails to converge.
What Differentiability Looks Like on a Graph
Geometrically, a function is differentiable at a point when you can draw exactly one tangent line there with a well-defined, finite slope. Zoom in far enough on the curve at a differentiable point, and it starts to look like a straight line. That “locally straight” quality is the visual signature of differentiability.
At a corner, zooming in never smooths out the sharp angle. At a cusp, the curve pinches to a point no matter how close you look. At a vertical tangent, the curve keeps looking vertical. These are all cases where the tangent line either doesn’t exist as a single line or exists but has no finite slope.
How the Idea Extends to Multiple Variables
For functions of two or more variables, the concept generalizes in an important way. A function f(x, y) is differentiable at a point if there’s a flat plane (called the tangent plane) that approximates the function well near that point. “Well” has a precise meaning: the gap between the function’s actual value and the plane’s value shrinks faster than the distance to the point itself.
Having partial derivatives in each variable separately is not enough. A function can have both a partial derivative with respect to x and a partial derivative with respect to y at a point yet still fail to be differentiable, because the function might behave badly along some diagonal approach. Differentiability in multiple dimensions requires the function to be well-approximated by a linear function from every direction simultaneously, not just along the coordinate axes.
A sufficient (though not necessary) condition that’s easy to check: if all partial derivatives exist and are continuous in a neighborhood of the point, the function is differentiable there.
Why Differentiability Matters in Practice
Differentiability isn’t just an abstract requirement. It’s the mathematical property that makes rates of change meaningful. When you compute velocity from a position function, you’re taking a derivative. If the position function isn’t differentiable at some moment in time, the velocity is undefined there, which would mean the object has no well-defined speed at that instant.
The same logic extends to any field that relies on rates of change. In economics, a differentiable cost function lets you calculate marginal cost. In engineering, differentiable stress-strain curves let you find the exact point where a material begins to yield. In optimization, the first step in finding a minimum or maximum is setting the derivative to zero, which only works if the derivative exists. Differentiability is the gateway to nearly all of calculus’s practical tools.