# 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.

what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
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how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
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