![]() The function is not continuous at this point. ![]() The function is continuous at this point since the function and limit have the same value. Jump discontinuities occur where the graph has a break in it as this graph does and the values of the function to either side of the break are finite (i.e. This kind of discontinuity in a graph is called a jump discontinuity. The function value and the limit aren’t the same and so the function is not continuous at this 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. 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. Let’s take a look at an example to help us understand just what it means for a function to be continuous.Įxample 1 Given the graph of f(x), shown below, determine if f(x) is continuous at x=−2, x =0, and x=3. ![]() 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. 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. This definition can be turned around into the following fact. If either of these do not exist the function will not be continuous at x=a. Note that this definition is also implicitly assuming that both exist. It’s now time to formally define what we mean by “nice enough”.Ī function f(x) is said to be continuous at x=a ifĪ function is said to be continuous on the interval if it is continuous at each point in the interval. 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.
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