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The Calculus Lifesaver by Adrian Banner

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CHAPTER 5 ______________________________

Continuity and Differentiability

In general, there’s only one special thing about the graph of a function: it just has to obey the vertical line test. That’s not particularly exclusive. The graph could be all over the place—a little bit here, a vertical asymptote there, or any number of individual disconnected points wherever the hell they feel like being. So now we’re going to see what happens if we’re a little more exclusive: we want to look at two types of smoothness. First, continuity: intuitively, this means that the graph now has to be drawn in one piece, without taking the pen off the page. Second, differentiability: the intuition here is that there are no sharp corners in the graph. In both cases, ...

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