Excellent question Greg. Most functions are continuous everywhere, except at the 4 types of discontinuities. These are vertical asymptotes, jumps, removable discontinuities, and certain types of oscillation problems. You simply have to test for continuity at any point that might be in question. We have a formula for continuity at a point. It states if that the limit at a point is equal to the value of the function at the point, then the function is continuous there. If either side of the limit doesn't exist or if the limit doesn't equal the function at the point in question, the function will not be continuous there. Just like when we are looking at domains, we will look for the points where a function may not be defined, so also with continuity, we look for places where continuity may be "destroyed" or more pleasantly, "affected". Don't forget to check at the places where a piece-wise function changes definition. Discontinuities could occur there even if both branches are continuous in themselves. Hope that helps!
Excellent question Greg. Most functions are continuous everywhere, except at the 4 types of discontinuities. These are vertical asymptotes, jumps, removable discontinuities, and certain types of oscillation problems. You simply have to test for continuity at any point that might be in question. We have a formula for continuity at a point. It states if that the limit at a point is equal to the value of the function at the point, then the function is continuous there. If either side of the limit doesn't exist or if the limit doesn't equal the function at the point in question, the function will not be continuous there. Just like when we are looking at domains, we will look for the points where a function may not be defined, so also with continuity, we look for places where continuity may be "destroyed" or more pleasantly, "affected". Don't forget to check at the places where a piece-wise function changes definition. Discontinuities could occur there even if both branches are continuous in themselves. Hope that helps!
ReplyDeleteBTW, let me know if there is a specific problem you have a question on and I can use it as an example.
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