A step function, also known as a simple function, is a finite sum of characteristic functions of bounded intervals. They are often used in real analysis and measure theory to approximate integrable functions.
Let $\{I_k\}_{k=1}^{n}$ be a finite set of bounded intervals. A corresponding step function $S:\mathbb{R} \to \mathbb{R}$ is a function of the form $$ S(x) = \sum_{k=1}^{n} a_k \Large{\chi}_{I_k}(x) $$Special cases include the sign function and the Heaviside theta function. The Kronecker delta is not typically taken to be a step function as the intervals are required to have positive length.
Step functions are continuous except possibly at boundary points of the intervals and are integrable. They often form a 'first step' for proving measurability properties of functions: first one might show the result for step functions, then measurable functions, then continuous functions, etc.
