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In this section, we will study continuous functions.

Functions that don't have breaks or jumps in their graphs are continuous.

The formal definition of a continuous function corresponds closely with the meaning of continuity in everyday language.

A function f is continuous at the point a if a is in the domain of f, exists, and . So, a continuous function has the property that a small change in the input produces only a small change in the output.

To decide whether a function is continuous on an interval, we must consider endpoints and interior points separately.

First, let's study the definition of continuity at the endpoints of an interval.

A function is continuous at the left endpoint a if .

A function is continuous at the right endpoint b if .

The endpoints are defined separately because they can only be checked for continuity from one direction. If the limit of an endpoint is checked from the side that is not in the domain, the values will not be in the domain and won't apply to the function.

Now, let's see whether is continuous on the closed interval [-1,1].

First, we check points on the interior of the interval. That is, we check for continuity at x-values between -1 and 1.

To see if f is continuous at these values, we need to check whether for all points between -1 and 1.

If a is between -1 and 1, we can use limit laws to show that is equal to . Because a is in the domain of 1 - x2, we can substitute a for x to find . So, .

Now, notice that is also equal to f(a). So, . Therefore, f is continuous at every point between -1 and 1.

To check for continuity at the endpoints, we must check the limit of f(x) as x approaches -1 from the left and 1 from the right.

Using similar calculations, we can see that , which is equal to f(-1), and , which is equal to f(1). So f is continuous at 1 and -1. Because f is continuous at the endpoints and all interior points, we can say that f is continuous on the closed interval [-1,1].

The graph on the left represents a function that is continuous at point a because it satisfies all three conditions. However, the function represented by the graph on the right is not continuous at a because does not exist. We say that this function is discontinuous at a.

Consider the function . This function is discontinuous at x = -2 because -2 is not in the domain of g(x). So, condition 1 does not hold.

The piecewise function g(x) defined to the right is not continuous at -2. Which of the three conditions for continuity are not satisfied at x = -2? Click "Submit" when finished.

Consider the function . Let's determine whether this function is continuous on the closed interval [-1,1].

Copyright 2006 The Regents of the University of California and Monterey Institute for Technology and Education