Unizor - Derivatives - Function Limit - Continuity

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),
{x₁,y₁=f(x₁)} and
{x₂,y₂=f(x₂)},
if x₁ is close to x₂,
then y₁ will be close to y₂.
And the "closeness" should be understood in a sense ofinfinitesimal distance, that is,
if x₁→x₂ then f(x₁)→f(x₂).

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 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 that
if x→r, then f(x)→f(a)

The rule (2) above means, usingε-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 f(x)=x³ is continuous.

Proof

This function is defined on a contiguous interval (−∞,+∞).
Choose any real number r and any positive ε.
Notice that
|x³−r³| = |x−r|·|x²+xr+r²| ≤
≤ |x−r|·(|x²|+|x|·|r|+|r²|)

To make the right side smaller than ε it is sufficient to chooseδ smaller than the minimum among δ₁=|r| (in which case |x|is not greater than 2|r| and the expression in parenthesis is not greater than 7r²) and δ₂=ε/(7r²)(in which case an entire right hand side of this inequality is not greater than ε).

2. Prove that f(x)=sin(x) is continuous.

Proof

This function is defined on a contiguous interval (−∞,+∞).
Choose any real number r and any positive ε.
Notice that
|sin(x)−sin(r)| =
= 2|sin((x−r)/2)·cos((x+r)/2)| ≤
≤ |x−r|

So, to make |sin(x)−sin(r)|smaller than ε, it is sufficient to chose δ=ε.

3. Let's define the following function:
f(x)=0 for all real x, except x=0and f(x)=1 for x=0.
Prove that it is not continuous.

Proof

This function is defined on a contiguous interval (−∞,+∞). So, the first requirement ofcontinuity is satisfied.
The second requirement, however, is not satisfied forr=0.
Indeed, the function f(x) has limit 0 when x approaching point 0 because all its values outside of point x=0 are equal to 0, but f(0)=1.
So, f(x) does not tend to f(0) as x→0.