2.4 Continuity  (Page 29/16)

Many functions have the property that their graphs can be traced with a pencil without lifting the pencil from the page. Such functions are called continuous . Other functions have points at which a break in the graph occurs, but satisfy this property over intervals contained in their domains. They are continuous on these intervals and are said to have a discontinuity at a point where a break occurs.

We begin our investigation of continuity by exploring what it means for a function to have continuity at a point . Intuitively, a function is continuous at a particular point if there is no break in its graph at that point.

Continuity at a point

Before we look at a formal definition of what it means for a function to be continuous at a point, let’s consider various functions that fail to meet our intuitive notion of what it means to be continuous at a point. We then create a list of conditions that prevent such failures.

Our first function of interest is shown in [link] . We see that the graph of $f\left(x\right)$ has a hole at a . In fact, $f\left(a\right)$ is undefined. At the very least, for $f\left(x\right)$ to be continuous at a , we need the following condition:

However, as we see in [link] , this condition alone is insufficient to guarantee continuity at the point a . Although $f\left(a\right)$ is defined, the function has a gap at a . In this example, the gap exists because $\underset{x\to a}{\text{lim}}f\left(x\right)$ does not exist. We must add another condition for continuity at a —namely,

However, as we see in [link] , these two conditions by themselves do not guarantee continuity at a point. The function in this figure satisfies both of our first two conditions, but is still not continuous at a . We must add a third condition to our list:

Now we put our list of conditions together and form a definition of continuity at a point.

The following procedure can be used to analyze the continuity of a function at a point using this definition.