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Continuity

# Continuity - Continuity Over the last few sections weve...

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Continuity Over the last few sections we’ve been using the term “nice enough” to define those functions that we could evaluate limits by just evaluating the function at the point in question. It’s now time to formally define what we mean by “nice enough”. Definition A function is said to be continuous at if A function is said to be continuous on the interval [ a, b ] if it is continuous at each point in the interval. This definition can be turned around into the following fact. Fact 1 If is continuous at then, This is exactly the same fact that we first put down back when we started looking at limits with the exception that we have replaced the phrase “nice enough” with continuous. It’s nice to finally know what we mean by “nice enough”, however, the definition doesn’t really tell us just what it means for a function to be continuous. Let’s take a look at an example to help us understand just what it means for a function to be continuous. Example 1 Given the graph of f(x) , shown below, determine if f(x) is continuous at , , and .

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Solution To answer the question for each point we’ll need to get both the limit at that point and the function value at that point. If they are equal the function is continuous at that point and if they aren’t equal the function isn’t continuous at that point. First . The function value and the limit aren’t the same and so the function is not continuous at this point. This kind of discontinuity in a graph is called a jump discontinuity . Jump discontinuities occur where the graph has a break in it is as this graph does. Now . The function is continuous at this point since the function and limit have the same value. Finally . The function is not continuous at this point. This kind of discontinuity is called a removable discontinuity . Removable discontinuities are those where there is a hole in the graph as there is in this case. From this example we can get a quick “working” definition of continuity. A function
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Continuity - Continuity Over the last few sections weve...

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