5.1 Integrals of step functions

We begin by defining the integral of certain (but not all) bounded, real-valued functions whose domains are closed bounded intervals.Later, we will extend the definition of integral to certain kinds of unbounded complex-valued functionswhose domains are still intervals, but which need not be either closed or bounded.

We begin by defining the integral of certain (but not all) bounded, real-valued functions whose domains are closed bounded intervals.Later, we will extend the definition of integral to certain kinds of unbounded complex-valued functionswhose domains are still intervals, but which need not be either closed or bounded.First, we recall from [link] the following definitions.

Let $\left[a,b\right]$ be a closed bounded interval of real numbers. By a partition of $\left[a,b\right]$ we mean a finite set $P=\left\{{x}_{0}<{x}_{1}<...<{x}_{n}\right\}$ of $n+1$ points, where ${x}_{0}=a$ and ${x}_{n}=b.$

The $n$ intervals $\left\{\left[{x}_{i-1},{x}_{i}\right]\right\}$ are called the closed subintervals of the partition $P,$ and the $n$ intervals $\left\{\left({x}_{i-1},{x}_{i}\right)\right\}$ are called the open subintervals or elements of $P.$

We write $\parallel P\parallel$ for the maximum of the numbers (lengths of the subintervals) $\left\{{x}_{i}-{x}_{i-1}\right\},$ and call $\parallel P\parallel$ the mesh size of the partition $P.$

If a partition $P=\left\{{x}_{i}\right\}$ is contained in another partition $Q=\left\{{y}_{j}\right\},$ i.e., each ${x}_{i}$ equals some ${y}_{j},$ then we say that $Q$ is finer than $P.$

Let $f$ be a function on an interval $\left[a,b\right],$ and let $P=\left\{{x}_{0}<...<{x}_{n}\right\}$ be a partition of $\left[a,b\right].$ Physicists often consider sums of the form

${S}_{P,\left\{{y}_{i}\right\}}=\sum _{i=1}^{n}f\left({y}_{i}\right)\left({x}_{i}-{x}_{i-1}\right),$

where ${y}_{i}$ is a point in the subinterval $\left({x}_{i-1},{x}_{i}\right).$ These sums (called Riemann sums) are approximations of physical quantities, and the limit of these sums, as the mesh of the partition becomes smaller and smaller,should represent a precise value of the physical quantity. What precisely is meant by the “ limit” of such sums is already a subtle question,but even having decided on what that definition should be, it is as important and difficult to determine whether or not such a limit exists for many (or even any) functions $f.$ We approach this question from a slightly different point of view, but we will revisit Riemann sums in the end.

Again we recall from [link] the following.

Let $\left[a,b\right]$ be a closed bounded interval in $R.$ A real-valued function $h:\left[a,b\right]\to R$ is called a step function if there exists a partition $P=\left\{{x}_{0}<{x}_{1}<...<{x}_{n}\right\}$ of $\left[a,b\right]$ such that for each $1\le i\le n$ there exists a number ${a}_{i}$ such that $h\left(x\right)={a}_{i}$ for all $x\in \left({x}_{i-1},{x}_{i}\right).$

REMARK A step function $h$ is constant on the open subintervals (or elements) of a certain partition. Of course, the partition is not unique.Indeed, if $P$ is such a partition, we may add more points to it, making a larger partition having moresubintervals, and the function $h$ will still be constant on these new open subintervals. That is, a given step function can be described using various distinct partitions.

Also, the values of a step function at the partition points themselves is irrelevant. We only require that it be constant on the open subintervals.

Let $h$ be a step function on $\left[a,b\right],$ and let $P=\left\{{x}_{0}<{x}_{1}<...<{x}_{n}\right\}$ be a partition of $\left[a,b\right]$ such that $h\left(x\right)={a}_{i}$ on the subinterval $\left({x}_{i-1},{x}_{i}\right)$ determined by $P.$

1. Prove that the range of $h$ is a finite set. What is an upper bound on the cardinality of this range?
2. Prove that $h$ is differentiable at all but a finite number of points in $\left[a,b\right].$ What is the value of ${h}^{\text{'}}$ at such a point?
3. Let $f$ be a function on $\left[a,b\right].$ Prove that $f$ is a step function if and only if ${f}^{\text{'}}\left(x\right)$ exists and $=0$ for every $x\in \left(a,b\right)$ except possibly for a finite number of points.
4. What can be said about the values of $h$ at the endpoints $\left\{{x}_{i}\right\}$ of the subintervals of $P?$
5. (e) Let $h$ be a step function on $\left[a,b\right],$ and let $j$ be a function on $\left[a,b\right]$ for which $h\left(x\right)=j\left(x\right)$ for all $x\in \left[a,b\right]$ except for one point $c.$ Show that $j$ is also a step function.
6. If $k$ is a function on $\left[a,b\right]$ that agrees with a step function $h$ except at a finite number of points ${c}_{1},{c}_{2},...,{c}_{N},$ show that $k$ is also a step function.

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