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Differentiable functions are Continuous

Differentiable functions are Continuous. Connecting Differentiability and Continuity. Differentiability and Continuity. Continuous functions that are not necessarily differentiable. (E.g. )

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Differentiable functions are Continuous

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  1. Differentiable functions are Continuous Connecting Differentiability and Continuity

  2. Differentiability and Continuity Continuous functions that are not necessarily differentiable. (E.g. ) If a function is differentiable we know that “if we zoom in sufficiently far” we will see a straight line. Our intuition thus tells us that locally linear functions cannot have “breaks in the graph. How do we prove this?

  3. But first . . . a preliminary idea (a +h, f(a + h)) (x, f(x)) (a, f(a)) (a, f(a)) a a x a + h Same picture, different labeling! These are just different ways of expressing the same mathematical idea!

  4. Differentiable Functions are Continuous Suppose that f is differentiable at x = a. Notice that: Is an alternate way of defining the derivative of f at x = a.

  5. Differentiable Functions are Continuous • In the end, this tells us that: Which is what it means to say that f is continuous at a ! So if f is differentiable at x = a, then f must also be continuous at x = a

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