# Calculus Of One Real Variable – By Pheng Kim Ving Chapter 2: The Derivative – Section 2.4: Differentiability Vs Continuity

2.4
Differentiability Vs Continuity

 1. Differentiability Implies Continuity

We'll show that if a function is differentiable, then it's continuous.

Theorem 1.1

 If a function f is differentiable at a point x = a, then f is continuous at x = a.

Proof EOP

Note that if we let h = xa then: The right-hand side of the above equation looks more familiar: it's used in the definition of the derivative.

 2. Continuity Doesn't Imply Differentiability

We'll show by an example that if f is continuous at x = a, then f may or may not be differentiable at x = a. The converse
to the above theorem isn't true. Continuity doesn't imply differentiability.

Example 2.1 Solution
a. Fig. 2.1  and thus f '(0) don't exist. It follows that f is not differentiable at x = 0.

Remark 2.1

In handling continuity and differentiability of f, we treat the point x = 0 separately from all other points because f changes
its formula at that point. We do so because continuity and differentiability involve limits, and when
f changes its formula at
a point, we must investigate the one-sided limits at both sides of the point to draw the conclusion about the limit at that
point.

 3. Where Functions Aren't Differentiable  Fig. 3.1   f isn't differentiable at a where it's discontinuous, at b where its graph has a sharp point, and at c where its graph has a vertical tangent line.

## Problems & Solutions

1. Let y = f(x) = x1/3.
a.  Sketch a graph of
f using graphing technology.
b.  Based on the graph, where is
f both continuous and differentiable?
c.  Based on the graph, where is
f continuous but not differentiable?

Solution

a. b.  Based on the graph, f is both continuous and differentiable everywhere except at x = 0.

c.  Based on the graph, f is continuous but not differentiable at x = 0. 2.  Let f be defined by f(x) = |x2 + 2x3|.
a.
Show that f is continuous everywhere.
b.  Show, using the definition of derivative, that
f is differentiable everywhere except at x =3 and x = 1.

Solution

a.  We have f(x) = |(x + 3)(x1)|. The following table shows the signs of (x + 3)(x1). So we have: Similarly, f is also continuous at x = 1. It follows that f is continuous everywhere.

 b. Case Where x < – 3 Or x > 1.  We have: Case Where – 3 < x < 1.  We have: So f is differentiable on (– 3, 1).

Case Where x = – 3.  We have: and thus f '(– 3) don't exist. As a consequence, f isn't differentiable at x = – 3.

Case Where x = 1.  Similarly, f isn't differentiable at x = 1.

In summary, f is differentiable everywhere except at x = – 3 and x = 1.  Solution  Note

Many other examples are possible, as seen in the figure below.   Solution   5.  If possible, give an example of a differentiable function that isn't continuous.

Solution

That's impossible, because if a function is differentiable, then it must be continuous.