A function is differentiable at a point when its derivative exists there — that is, when the difference-quotient limit \lim_{h \to 0}\tfrac{f(x+h)-f(x)}{h} settles on one finite value. Geometrically: the curve has a single, well-defined tangent line at that point.

Most of the smooth curves we have met are differentiable everywhere. But some points break the rule, and it is worth knowing exactly where and why.

Differentiable ⇒ continuous

There is a one-way street between two ideas. If f is differentiable at a point, then it must be continuous there:

The intuition: to have a tangent, the curve can't jump. A break in the graph means the nearby points fly off in a wildly different direction depending on which side you approach from, so no single slope exists.

The arrow runs only one way. Continuous does not imply differentiable — a curve can be perfectly connected yet still fail to have a tangent at a sharp spot. The next card shows the three classic ways that happens.

Three ways to fail: corner, cusp, vertical tangent

Each of these graphs is continuous — you could draw it without lifting your pen — yet none has a derivative at the marked point. Step through them.

• Corner. f(x) = |x| at x = 0: the slope is -1 from the left and +1 from the right. The two don't agree, so the limit fails.
• Cusp. f(x) = x^{2/3} at x = 0: the curve comes to a sharp point where the slopes shoot to -\infty and +\infty.
• Vertical tangent. f(x) = x^{1/3} at x = 0: the tangent is straight up, so its slope is infinite — not a finite number.

Zoom in: smooth straightens, the corner doesn't

A differentiable curve looks straighter the closer you zoom — it is "locally linear", which is what lets it have a tangent. A corner never straightens. The faint curve is the smooth f(x) = x^2; the bold one is the corner g(x) = |x|. Slide the zoom toward 0 and watch only the parabola flatten into a line — the kink in |x| stays exactly as sharp.

Watch it explained

Sal Khan connects differentiability and continuity, and shows the places a derivative fails to exist.