1. Differentiable functions are Continuous
Connecting Differentiability
and Continuity

2. Differentiability and Continuity
Continuous functions are not necessarily differentiable.
For instance, start with

3. Differentiability and Continuity
Continuous functions are not necessarily differentiable.
. . . Now take absolute values

4. Differentiability and Continuity
Continuous functions are not necessarily differentiable.
. . . Now take absolute values

5. Differentiability and Continuity
Continuous functions are not necessarily differentiable. (E.g )
A function is differentiable “if we see a straight line when we zoom in sufficiently far.”

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

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

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

9. Differentiable Functions are Continuous
Suppose that f is differentiable at x = a. Then
In order to show that f is continuous at x = a, we have to show that

10. Differentiable Functions are Continuous
Suppose that f is differentiable at x = a. Then we know the limit
of the difference quotient exists and is equal to

11. 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