Differentiable functions are Continuous Connecting Differentiability and Continuity

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?

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

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.

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