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

 

2.4
Differentiability Vs Continuity

 

 

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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 = x – a then:

 

 

The right-hand side of the above equation looks more familiar: it's used in the definition of the derivative.

 

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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.

 

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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.

 

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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.

 

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2.  Let f be defined by f(x) = |x2 + 2x – 3|.
     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)(x – 1)|. The following table shows the signs of (x + 3)(x – 1).

 

   

 

    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.

 

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Solution

 

 

 

Note

 

Many other examples are possible, as seen in the figure below.

 

 

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Solution

 

 

 

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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.

 

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