Smooth Is a Stronger Word Than Unbroken

You'll learnwhy a graph with no break can still have no slope at a point, which way the one true implication runs, and how to name the four ways a derivative fails.

Here is a graph you can draw without lifting the pen: a V. It has no gap, no jump, nothing missing. Does it have a slope at the bottom of the V? Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · DIFFERENTIABLE ⇒ CONTINUOUS, NEVER THE REVERSE · CORNER · CUSP · VERTICAL TANGENT · BREAK · v2

differentiable ⇒ continuous · the reverse arrow is false

Before you start

What you're looking at

A V-shaped graph with no break in it, and the point at the bottom of the V.

The question

If a graph has no break at a point, is that enough for it to have a slope there, and if not, what goes wrong?

Step 1 / 12