Characteristics that destroy differentiability

Sharp Turns (right-hand limit does not equal left-hand limit)

Graph y = |x|
This function is not differentiable at x = 0

Vertical Tangents

Graph y = x^1/3
This function is not differentiable at x = 0 because the slope does not exist there.

Discontinuities

Graph y = 1/x
This function is not differentiable at x = 0 because the function does not exist there.

Differentiability implies local linearity
If a function is differentiable at a point, then as you zoom in on that point , eventually all you will
see on your screen is a straight line. This is a very close approximation to the tangent line at that
point. If a function is not differentiable, this will not happen.

Differentiability implies continuity

Given: f(x) is differentiable. Therefore,

lim f(x) - f(c) exists at x = c
x®c x - c

Prove: f(x) is continuous. (Use the 3-step proof)

PROOF:

i) f(c) exists because the given limit exists and f(c) must be used to calculate the limit

ii) lim [f(x) - f(c)] = lim [f(x) - f(c)](x - c)
x®c x®c x - c

= lim [f(x) - f(c)] (x - c)
x®c x - c

= lim f(x) - f(c) lim (x - c)
x®c x - c

= [f ' (c)](0)

= 0

Since lim [f(x) - f(c)] = 0, then lim f(x) = f(c) and the limit exists
x®c x®c

iii) lim f(x) = f(c)
x®c

Therefore, f(x) is continuous at x = c

True /False Statements:

Differentiability implies continuity TRUE
Continuity implies differentiability FALSE (converse statement)

Consider functions whose graphs have sharp points

Not continuous implies not differentiable TRUE (contrapositive statement)
Not differentiable implies not continuous FALSE (inverse statement)

Again,consider functions whose graphs have sharp points

If a statement is true, its contrapositive is always true

Problems