# Differentiability

A function is differentiable at a if its derivative exists there. A sharp corner, a vertical tangent or a break means it is not.

Maths (university) · Part 03: Differentiation · Card 10 of 10 · Last updated 2026-10-19

## Analogy

A smooth hill is easy to slope-check, but at a pointed roof peak there is no single slope.

## Example

**f(x) = |x| at x = 0**

Right of 0 the slope is 1.
Left of 0 the slope is -1.
The slopes disagree.
So f'(0) does not exist.

## In practice

Many theorems need differentiability, and every differentiable function is continuous.

## Practice

### Flashcards

- **Is |x| differentiable at 0?** No. The left slope is -1 and the right slope is 1.

## Questions people ask

### Does continuity imply differentiability?

No. The absolute value function is continuous at 0 but has a sharp corner there, so it has no derivative. The reverse holds: if a function is differentiable at a point, it must be continuous at that point. So continuity is necessary for a derivative, but it is not enough.

### How can I tell where a function is not differentiable?

Look for corners, cusps, jumps and vertical tangents. For |x|, the left and right slopes disagree at 0. For the cube root of x, the slope is infinite at 0. Algebraically, check that the one-sided derivatives exist and are equal. If they differ, or either is infinite, the derivative does not exist at that point.

## Related concepts

- [Derivative](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-1)
- [Tangent line](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-2)
- [Rate of change](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-3)
- [Power rule](https://studyvisually.com/learn/science/maths-university/maths-university-part-3/maths-university-part-3-concept-4)

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