# Limits and continuous functions

## Limits

### Limits of real functions

#### Limit operator

For a function \(f(x)\),

\(\lim_{x\rightarrow a}f(x)=L\)

We can say that \(L\) is the limit if:

\(\forall \epsilon >0 \exists \delta >0 \forall x[0<|x-p|<\delta \rightarrow |f(x)-L|<\epsilon]\)

### Limit superior and limit inferior

If a sequence does not converge, but stays between two points, then lim sup is upper bound, lim inf is lower bound.

## Continuous functions

### Continous functions

A function is continuous if:

\(\lim_{x\rightarrow c} f(x)=f(c)\)

For example a function \(\dfrac{1}{x}\) is not continuous as the limit towards \(0\) is negative infinity. A function like \(y=x\) is continous.

More strictly, for any \(\epsilon >0\) there exists

\(\delta >0 \)

\(c-\delta < x< c +\delta \)

Such that

\(f(c)-\epsilon < f(x) < f(c)+\epsilon \)

This means that our function is continuous at our limit \(c\), if for any tiny range around \(f(c)\), that is \(f(c)-\epsilon\) and \(f(c)+\epsilon\), there is a range around \(c\), that is \(c-\delta \) and \(c+ \delta \) such that all the value of \(f(x)\) at all of these points is within the other range.

#### Limits

Why can’t we use rationals for analysis?

If discontinous at not rational number, it can still be continous for all rationals.

Eg \(f(x)=-1\) unless \(x^2>2\), where \(f(x)=1\).

Continous for all rationals, because rationals dense in reals.

But can’t be differentiated.

### Reals or rationals for analysis

Why can’t we use rationals for analysis?

If discontinous at not rational number, it can still be continous for all rationals.

eg \(f(x)=-1\) unless \(x^2>2\), where \(f(x)=1\).

Continous for all rationals, because rationals dense in reals

But can’t be differentiated

### Boundedness theorem

If \(f(x)\) is closed and continuous in \([a,b]\) then \(f(x)\) is bounded by \(m\) and \(M\). That is:

\(\exists m \in \mathbb{R} \exists M\in \mathbb{R}\forall x\in [a,b](m<f(x)<M\)

Take a real function \(f(x)\) on closed interval \([a,b]\), continuous on \([a,b,]\).

IVT says that for all numbers \(u\) between \(f(a)\) and \(f(b)\), there is a corresponding value \(c\) in \([a,b]\) such that \(f(c)=u\).

That is:

\(\forall u \in [min(f(a),f(b)),max(f(a),f(b))] \exists c \in [a,b] (f(c)=u)\)

### Extreme value theorem

We can expand the boundedness theorem such that \(m\) and \(M\) are functions of \(f(x)\) in the bound \([a,b]\). That is:

\(\exists m \in \mathbb{R} \exists M\in \mathbb{R}\forall x\in [a,b](m<f(x)<M)\)