If f ( a ) f ( a ) is defined, continue to step 2. If f ( a ) f ( a ) is undefined, we need go no further. Check to see if f ( a ) f ( a ) is defined.Problem-Solving Strategy: Determining Continuity at a Point At the very least, for f ( x ) f ( x ) to be continuous at a, we need the following condition: We see that the graph of f ( x ) f ( x ) has a hole at a. Our first function of interest is shown in Figure 2.32. We then create a list of conditions that prevent such failures. Continuity at a Pointīefore 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.

Intuitively, a function is continuous at a particular point if there is no break in its graph at that point. We begin our investigation of continuity by exploring what it means for a function to have continuity at a point. They are continuous on these intervals and are said to have a discontinuity at a point where a break occurs. Other functions have points at which a break in the graph occurs, but satisfy this property over intervals contained in their domains. Many functions have the property that their graphs can be traced with a pencil without lifting the pencil from the page.

2.4.5 Provide an example of the intermediate value theorem.

2.4.4 State the theorem for limits of composite functions.

2.4.3 Define continuity on an interval.

2.4.2 Describe three kinds of discontinuities.

2.4.1 Explain the three conditions for continuity at a point.

Continuity calculus patrickjmt