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 + h a x
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