So far, we have proved three main results concerning limits and
arithmetical operations $+,\xb7,-$ and (with one important
proviso) a similar result for divison. Because of the
algebraic

nature of addition, multiplication and division, such
results are often grouped together under the title The Algebra Of
Limits

. I find this name very unfortunate for a number of reasons:
first, it suggests the work here has been algebraic (it hasn't: it has
been analytic); second, it suggests that the results are
true *because of* the special algebraic

nature of the
operations on limits being considered; third, the initials are "AOL".

To really understand what is going on here we need to take a slightly
more abstract view and look to see what addition, subtraction and
multiplication have in common. These are all functions, defined on the
real numbers, with two inputs

or arguments. If $f(x,y)$
is the function $x+y$, or $x-y$, or $x\xb7y$ then
$f:{\mathbb{R}}^{2}\to \mathbb{R}$ is
a function with the following property. For all sequences $\left({a}_{n}\right)$
and $\left({b}_{n}\right)$ we have,

Where, by this we mean: *if the limits on the right hand side
exist in the reals then the limit on the left exists and equals the
expression given*. This is the main content of the last two web pages
on sums of sequences and
products of sequences.

Other functions have this property such as the distance function
$d(x,y)=\left|x-y\right|$. In fact there is no need for the function
to have exactly two arguments, and the idea works just as well
with one, two, three or more arguments. Some functions do not have
the property though. Consider the integer part

function,
$\left[x\right]$. If we define a sequence
$\left({a}_{n}\right)$ by ${a}_{n}=1-\frac{1}{n}$ then $\underset{n\to \infty}{\text{lim}}{a}_{n}=1$ by the convergence of the constant sequence $1$ to $1$,
the convergence of $\frac{1}{n}$ to $0$, and the result
mentioned above on subtraction, that $\underset{n\to \infty}{\text{lim}}{b}_{n}-{c}_{n}=\underset{n\to \infty}{\text{lim}}{b}_{n}-\underset{n\to \infty}{\text{lim}}{c}_{n}$ where ${b}_{n}=1$ and ${c}_{n}=1/n$. But $\left[{a}_{n}\right]=[1-\frac{1}{n}]=0$ for all $n$, so $\underset{n\to \infty}{\text{lim}}\left[{a}_{n}\right]=0\ne \left[\underset{n\to \infty}{\text{lim}}{a}_{n}\right]$. In other words, limits cannot be pushed through

integer-part
signs. Examining the graph of $y=\left[x\right]$ gives some intuition as
to the reason. This graph has a jump

at $x=1$. In other words
it is not *continuous*.

Definition.

Let $f:\mathbb{R}\to \mathbb{R}$ be a function and $l\in \mathbb{R}$. We say that $f$ is continuous at $l$ if whenever $\left({a}_{n}\right)$ is a sequence that converges to $l$ then the sequence $\left({b}_{n}\right)$ defined by ${b}_{n}=f\left({a}_{n}\right)$ also converges and its limit is equal to $f\left(l\right)$. We say that $f$ is continuous if it is continuous at all $l\in \mathbb{R}$.

So for example, the function $f\left(x\right)=\left[x\right]$ is not continuous at $l$ for $l=\dots ,-2,-1,0,1,2,\dots $ but the functions $g\left(x\right)=2x$ and $f\left(x\right)={x}^{2}$ are continuous at all values of $l$ since $\underset{n\to \infty}{\text{lim}}2{a}_{n}=2\left(\underset{n\to \infty}{lim}{a}_{n}\right)$ and $\underset{n\to \infty}{\text{lim}}{{a}_{n}}^{2}={\left(\underset{n\to \infty}{\text{lim}}{a}_{n}\right)}^{2}$ for all convergent sequences $\left({a}_{n}\right)$.

Definition.

Let $f:{\mathbb{R}}^{2}\to \mathbb{R}$ be a function with two arguments and $l,m\in \mathbb{R}$. We say that $f$ is continuous at $(l,m)$ if whenever $\left({a}_{n}\right)$ and $\left({b}_{n}\right)$ are sequences that converge to $l$, $m$ respectively then the sequence $\left({c}_{n}\right)$ defined by ${c}_{n}=f({a}_{n},{b}_{n})$ also converges and its limit is equal to $f(l,m)$. We say that $f$ is continuous if it is continuous at all $(l,m)\in {\mathbb{R}}^{2}$.

Continuity is about pushing functions through limits

and the most important continuous functions are the arithmetic
ones such as plus and times. The following theorem covers the
most important cases of these:

Continuity of plus, times, etc.

- The addition function $+$ with two real arguments is continuous.
- The subtraction function $-$ with two real arguments is continuous.
- The multiplication function $\xb7$ with two real arguments is continuous.
- The division function $/$ (i.e., the function defined by $f(x,y)=x/y$) with two real arguments is continuous at all $(l,m)\in {\mathbb{R}}^{2}$ except when $m=0$.
- The square root function $\sqrt{\text{}}$ is continuous at all positive $l\in \mathbb{R}$.
- The absolute value function $\left|\right|$ is continuous.

The first four parts of this have already been proved. We sketch the proof of the others here.

Proposition.

The square root function $\sqrt{x}$ is continuous at all $l>0$.

**Proof.**

Let ${a}_{n}\to l$ as $n\to \infty $, where the limit $l$ is positive. The by a previous proposition there is ${N}_{0}\in \mathbb{N}$ such that ${a}_{n}>\left|l\right|/2$ for all $n\u2a7e{N}_{0}$. Thus the sequence $\left(\sqrt{{a}_{n}}\right)$ is meaningful. We show that this sequence converges to $\sqrt{l}$.

**Subproof.**

Let $\epsilon >0$ be arbitrary.

**Subproof.**

Let ${N}_{1}\in \mathbb{N}$ such that $\forall n\u2a7e{N}_{1}\left|{a}_{n}-l\right|\epsilon \sqrt{l}$.

Let $N=max({N}_{0},{N}_{1})$.

**Subproof.**

Let $n\u2a7eN$ be arbitrary.

Then ${a}_{n}>0$ and $\left|\sqrt{{a}_{n}}-\sqrt{l}\right|=\frac{\left|{a}_{n}-l\right|}{\sqrt{{a}_{n}}+\sqrt{l}}<\frac{\epsilon \sqrt{l}}{\sqrt{l}}=\epsilon $.

So $\forall n\u2a7eN\left|\sqrt{{a}_{n}}-\sqrt{l}\right|\epsilon $.

So $\exists N\in \mathbb{N}\forall n\u2a7eN\left|\sqrt{{a}_{n}}-\sqrt{l}\right|\epsilon $.

So $\forall \epsilon >0\exists N\in \mathbb{N}\forall n\u2a7eN\left|\sqrt{{a}_{n}}-\sqrt{l}\right|\epsilon $.

Proposition.

The absolute value function $\left|\right|$ is continuous at all $l\in \mathbb{R}$.

**Proof.**

We have to prove that ${a}_{n}\to l$ implies $\left|{a}_{n}\right|\to \left|l\right|$. We do this in the case when $l>0$. The other cases ($l<0$ and $l=0$) are easy variations left as exercises.

**Subproof.**

Let $\epsilon >0$ be arbitrary.

**Subproof.**

Let ${N}_{0}\in \mathbb{N}$ such that $\forall n\u2a7e{N}_{0}{a}_{n}l/2$.

Let ${N}_{1}\in \mathbb{N}$ such that $\forall n\u2a7eN\left|{a}_{n}-l\right|\epsilon $.

Let $N=max({N}_{0},{N}_{1})$.

**Subproof.**

Let $n\u2a7eN$ be arbitrary.

Then ${a}_{n}$ and $l$ are both positive and so $\left|{a}_{n}\right|={a}_{n}$, $\left|l\right|=l$, so $\left|\left|{a}_{n}\right|-\left|l\right|\right|=\left|{a}_{n}-l\right|<\epsilon $.

So $\forall n\u2a7eN\left|\left|{a}_{n}\right|-\left|l\right|\right|\epsilon $.

So $\exists N\in \mathbb{N}\forall n\u2a7eN\left|\left|{a}_{n}\right|-\left|l\right|\right|\epsilon $.

So $\forall \epsilon >0\exists N\in \mathbb{N}\forall n\u2a7eN\left|\left|{a}_{n}\right|-\left|l\right|\right|\epsilon $.

These results can be combined to give other continuous functions.

The distance function $d(x,y)=\left|x-y\right|$ is continuous at all $(l,m)\in {\mathbb{R}}^{2}$.

**Proof.**

Let ${a}_{n}\to l$ and ${b}_{n}\to m$. Then $d({a}_{n},{b}_{n})=\left|{a}_{n}-{b}_{n}\right|$. So, taking limits as $n\to \infty $,

$$limd({a}_{n},{b}_{n})=lim\left|{a}_{n}-{b}_{n}\right|=\left|lim\left({a}_{n}-{b}_{n}\right)\right|,$$as $\left|\right|$ is continuous, and this limit equals

$$\left|lim{a}_{n}-lim{b}_{n}\right|=d(lim{a}_{n},lim{b}_{n}),$$as$-$ is continuous.

Continuity of the familiar functions we see here is useful in computing limits of what would otherwise be rather scary-looking sequences.

Example.

Let $\left({a}_{n}\right)$ be defined by

We compute its limit as $n\to \infty $ as follows.

as ${n}^{-1/2}$, ${n}^{-1}$, ${n}^{-2}$, ${n}^{-3}$ all converge to $0$, as $+$ and scalar multiplication are continuous everywhere, as $\sqrt{\text{}}$ is continuous at $3$ and $4$ and division is continuous at $(\sqrt{3},\sqrt{4})$.

Example.

Let the sequence $\left({a}_{n}\right)$ be defined by ${a}_{1}={a}_{2}=1$, and

Let us suppose that ${a}_{n}\to l\ne 0$ as $n\to \infty $. Then $\left({a}_{n+1}\right)$ and $\left({a}_{n+2}\right)$ are subsequences of $\left({a}_{n}\right)$ so by our assumption should tend to the same limit $l$. By continuity of addition, scalar multiplication everywhere and division at $(5l,6l)$ (noting our assumption $l\ne 0$) we have

But ${a}_{n+2}\to l$ so by the uniqueness of limits we must have $l=\frac{5}{6}$.

The conclusion is that **if the sequence $\left({a}_{n}\right)$
converges to some $l\ne 0$ then $l=\frac{5}{6}$.**
Of course this assumption is a big one and sequences defined in this
sort of way need not converge at all. We therefore need to find
ways to prove that a sequence converges without necessarily finding
the limit: this is the subject of the next block of work.

Pushing limits

through functions is a natural idea and can
save an enormous amount of time in calculations and proofs. You
have seen in this web page how it can be justified for some functions
and not for others. The functions for which we can manipulate limits
in this way are called continuous. Sometimes a function is
continuous at some points but not others. The integer-part function
and the division function are examples of these.

Implicitly assuming a function is continuous and
pushing a limit through

is one of the most common sources of errors in a mathematical
argument. There are many fallacious arguments

that prove
that $0=1$ or some such absurdity that reply on assuming a
non-continuous function is continous to trick the reader. Please
always state in your own work reasons for such
arguments (and check the functions really are continuous).