# Continuity and Limits

## Contents

#### Continuity

Continuity can be defined conceptually in a few different ways. A function is continuous, for example, if its graph can be traced with a pen without lifting the pen from the page. A function is continuous if its graph is an unbroken curve; that is, the graph has no holes, gaps, or breaks. But terms like "unbroken curve" and "gaps" aren't technical mathematical terms and at best, only provide a reader with a description of continuity, not a definition.

The more formal definition of continuity is this: a function f (x) is continuous at a point x = a , if and only if the following three conditions are met. 1) f (a) is defined. 2) f (x) exists. 3) f (x) = f (a) . Otherwise, the function is discontinuous.

A function can be continuous at a point, continuous over a given interval, or continuous everywhere. We have already defined continuity at a given point. For a function to be continuous over an interval [a, b] , that function must be continuous at each point in the interval, as well as at both a and b . For a function to be continuous everywhere, it must be continuous for every real number.

Discontinuities in functions can be classified according to the reason that the function is discontinuous at a given point. If there exists a vertical asymptote at x = a for a function, that function is said to have an infinite discontinuity at x = a .

Figure %: The function f (x) = has an infinite discontinuity at x = 1 .
If f (x)≠ f (x) , then f (x) does not exist, which means that the second condition necessary for continuity is not met. Such a discontinuity is called a jump discontinuity, and looks like this:
Figure %: The function f (x) = - 1 for x < 0 , f (x) = 1 for x≥ 0 has a jump discontinuity at x = 0 .
If f (x)≠f (a) , the function has what is often called a point, removable, or gap discontinuity. The point (a, f (x)) is often called a hole.
Figure %: The function f (x) = has a point discontinuity at x = 1 .