Interval/Real Function/Step function from above and from below/Definition

Step function from above

Let ${}I$ denote a bounded interval, and let

$f\colon I\longrightarrow \mathbb {R}$

denote a function. Then a step function

$t\colon I\longrightarrow \mathbb {R}$

is called a step function from above for ${}f$, if ${}t(x)\geq f(x)$ holds for all ${}x\in I$. A step function

$s\colon I\longrightarrow \mathbb {R}$

is called a step function from below for ${}f$, if ${}s(x)\leq f(x)$ holds for all ${}x\in I$.