The gradient is infinitely steep at the points where there is a step in value. These arrows are called delta functions. They are thought of as infinitely high and thin, and have a notional area equal to the value of the step jump. Thus when you integrate this derivative, you get back to the original step function.

A system's (for example a car suspension) response to an impulse (you might think of it as a sudden "bang") is very useful, and can be used on a process called "convolution" to find a system's response to any input.