*Notes to a video lecture on http://www.unizor.com*

__Continuous Functions__

A special type of functions is widely used in mathematics. They are called

*continuous functions*.

Simply speaking, the graph of these functions can be drawn in one movement of a pen without lifting it from the paper.

Our purpose is to define these functions more precisely, more rigorously, in order to use the term "

*continuous function*" wherever necessary without getting into details, properties and characteristics.

The action of drawing a graph without lifting a pen from paper implies, first of all, that the graph is a line. It's not necessarily straight, it might be curved, but it is contiguous, which means that the function is defined on some contiguous interval, finite or infinite to

*+∞*and/or

*−∞*.

We can say now that the

*domain*of a

*continuous function*is a contiguous interval, finite or infinite. Here are examples of the possible domains:

[

*a,b*]

(

*a,b*]

[

*a,+∞*)

(

*−∞,+∞*)

etc.

Let's now define the action of "not lifting a pen" mathematically.

In more precise terms it means that for two points on the graph of a function

**,**

*y=f(x)*{

*x*} and

_{1},y_{1}=f(x_{1}){

*x*},

_{2},y_{2}=f(x_{2})if

*x*is close to

_{1}*x*,

_{2}then

*y*will be close to

_{1}*y*.

_{2}And the "closeness" should be understood in a sense of

*infinitesimal*distance, that is,

if

*x*then

_{1}→x_{2}*f(x*.

_{1})→f(x_{2})It is important that this rule must be true for any point within the domain of a function, which leads us to the following more rigorous definition of a

*continuous function*.

**The real function**

(1) its domain is a contiguous interval, finite or infinite;

(2) for any point

if

*f(x)*is called continuous if(1) its domain is a contiguous interval, finite or infinite;

(2) for any point

*r*within its domain it is true thatif

*x→r*, then*f(x)→f(a)*The rule (2) above means, using

*ε-δ*language, that

for any positive

*ε*exists

*δ*such that,

if

**|**,

*x−r*| ≤*δ*then

**|**

*f(x)−f(r)*| ≤*ε**Problems*

1. Prove that

**is continuous.**

*f(x)=x³**Proof*

This function is defined on a contiguous interval (

*−∞,+∞*).

Choose any real number

**and any positive**

*r***.**

*ε*Notice that

**|**

*x³−r³*| = |*x−r*|·|*x²+xr+r²*| ≤**≤ |**

*x−r*|·(|*x²*|+|*x*|·|*r*|+|*r²*|)To make the right side smaller than

**it is sufficient to choose**

*ε***smaller than the minimum among**

*δ***(in which case**

*δ*=|_{1}*r*|**|**is not greater than

*x*|**2|**and the expression in parenthesis is not greater than

*r*|**) and**

*7r²***(in which case an entire right hand side of this inequality is not greater than**

*δ*_{2}=ε/(7r²)**).**

*ε*2. Prove that

**is continuous.**

*f(x)=sin(x)**Proof*

This function is defined on a contiguous interval (

*−∞,+∞*).

Choose any real number

**and any positive**

*r***.**

*ε*Notice that

**|**

= 2|

*sin(x)−sin(r)*| == 2|

*sin((x−r)/2)·cos((x+r)/2)*| ≤**≤ |**

*x−r*|So, to make

**|**smaller than

*sin(x)−sin(r)*|**, it is sufficient to chose**

*ε***.**

*δ=ε*3. Let's define the following function:

**for all real**

*f(x)=0***, except**

*x***and**

*x=0***for**

*f(x)=1***.**

*x=0*Prove that it is not continuous.

*Proof*

This function is defined on a contiguous interval (

*−∞,+∞*). So, the first requirement of

*continuity*is satisfied.

The second requirement, however, is not satisfied for

**.**

*r=0*Indeed, the function

**has limit**

*f(x)***when**

*0***approaching point**

*x***because all its values outside of point**

*0***are equal to**

*x=0***, but**

*0***.**

*f(0)=1*So,

**does not tend to**

*f(x)***as**

*f(0)***.**

*x→0*
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