Continuity

To clarify the concept of continuity, we can simply say that if a function is continuous on an interval, the graph of the function will not have any jumps or holes on that interval. Nevertheless, to speak precisely, that is not a satisfactory definition for mathematics. We can define the continuity of a function as following.

Definition

We say that a function f is continuous at a number a in its domain if and only if

Note that this definitions requires three conditions to be available:

1.      exists.

2.      exists.

3.     .

A function that does not satisfy any of these conditions at any point is said to be discontinuous at that point.

As there are one-sided limits, there is one-sided continuity, explained by the following definition.

Definition

A function f is continuous from the right at a number a in its domain if and only if

A function f is continuous from the left at a number a in its domain if and only if

We can deduce now that a function f is continuous at a point a if and only if

Corollary

All polynomial functions and the absolute value function are continuous at all real numbers. In addition, any rational function is continuous at all points in its domain. The principal square root function is continuous at all nonnegative numbers.

Theorem

If we have two functions, f and g, which are continuous at a number a, the functions f+g and fg are also continuous at a. If g(a) is not equal to zero, f/g is also continuous at a.

Theorem

If the function g is continuous at a number a, and f is continuous at g(a), the composite function  is also continuous at a.

Here are definitions for continuity on intervals.

Definition

A function is said to be continuous on an open interval if and only if it is continuous at every point in this interval.

Definition

A function f is said to be continuous on the closed interval  if and only if it is continuous on the open interval , continuous from the right at a and continuous from the left at b.

We conclude this section by the intermediate value theorem, which is easy to understand, but essential in proving many other theorems.

The Intermediate Value Theorem

If the function f is continuous on the closed interval  and if W is a number between f(a) and f(b), there exists a number c between a and b such that f(c)=W.

This theorem may explain how a continuous function can be graphed without lifting the pencil from the paper. In other words, it means that the function f which is continuous on  takes on every value between f(a) and f(b) as x represents different values between a and b.

Example

Examine the continuity of the function

at x=3.

Solution

Example

Examine the continuity of the function

at x=-5

Solution