What is a corner in calculus?

A derivative at a specified point is only defined for a function where there is only one slope at that specified point. A corner is one type of shape to a graph that has a different slope on either side. It is similar to a cusp.

You may see corners in the context of absolute value functions, like:

##y = -|x| + 2##: graph{ -|x| + 2 [-10, 10, -5, 5]}

Here, the derivative at ##x = 0## is undefined, because the slope on the left side is ##1##, but the slope on the right side is ##-1##.

Similarly, a cusp looks like this:

##y = 2sqrt(|x|)##: graph{2sqrt(|x|) [-10, 10, -5, 5]}

As you can see, it also has two different (and undefined) slopes at ##y = 0##, so it is considered to have an undefined derivative at that point.

If you're curious, the derivative of ##|x|## is ##x/|x|##.