DEVELOPMENT OF THE LESSON
(A) INTRODUCTION
The difference between continuity and differentiability is a critical issue. Most, but not all, of the functions we encounter in calculus will be differentiable over their entire domain.
Before we can confidently apply the rules regarding derivatives, we need to be able to recognize the exceptions to the rule.
Recall the following definitions:
Definition 1 (Continuity at a Number). A function f is continuous at a number c if all of the following conditions are satisfied:
(i) f(c) is defined;
(ii) lim
x!cf (x) exists; and (iii) lim
x!cf (x) = f (c).
If at least one of the these conditions is not satisfied, the function is said to be discontin-uous at c.
Definition 2(Continuity on R). A function f is said to be continuous everywhere if f is continuous at every real number.
Definition 3. A function f is differentiable at the number c if f0(c) = lim
h!0
f (c + h) f (c) h exists.
(B) LESSON PROPER
We now present several examples of determining whether a function is continuous or dif-ferentiable at a number.
EXAMPLE 1:
1. The piecewise function defined by
f (x) = 2. The function defined by
f (x) = 8<
:
x2 if x < 2, 3 x if x 2.
is not continuous at c = 2 since lim
x!2 f (x) = 4 6= 1 = lim
x!2+f (x), hence the lim
x!2f (x) does not exist.
3. Consider the function f(x) =p3
x. By definition, its derivative is f0(x) = lim does not exist. Hence f is not differentiable at x = 0.
4. The function defined by
f (x) =
Since the one-sided limits exist and are equal to each other, the limit exists and equals 5. So,
x!1limf (x) = 5 = f (1).
This shows that f is continuous at x = 1. On the other hand, computing for the derivative,
Since the one-sided limits at x = 1 do not coincide, the limit at x = 1 does not exist.
Since this limit is the definition of the derivative at x = 1, we conclude that f is not differentiable at x = 1.
5. Another classic example of a function that is continuous at a point but not differentiable at that point is the absolute value function f(x) = |x| at x = 0. Clearly, f(0) = 0 =
x!0lim|x|. However, if we look at the limit definition of the derivative,
h!0lim
Note that the absolute value function is defined differently to the left and right of 0 so we need to compute one-sided limits. Note that if h approaches 0 from the left, then it approaches 0 through negative values. Since h < 0 =) |h| = h, it follows that
hlim!0
Similarly, if h approaches 0 from the right, then h approaches 0 through positive values.
Since h > 0 =) |h| = h, we obtain
Hence, the derivative does not exist at x = 0 since the one-sided limits do not coincide.
The previous two examples prove that continuity does not necessarily imply differentiability.
That is, there are functions which are continuous at a point, but is not differentiable at that point. The next theorem however says that the converse is always TRUE.
Theorem 6. If a function f is differentiable at a, then f is continuous at a.
Proof. That function f is differentiable at a implies that f0(a) exists. To prove that f is continuous at a, we must show that
x!alimf (x) = f (a),
or equivalently,
hlim!0f (a + h) = f (a).
If h 6= 0, then
f (a + h) = f (a) + f (a + h) f (a)
= f (a) + f (a + h) f (a)
h · h.
Taking the limit as h ! 0, we get
h!0limf (a + h) = lim
h!0f (a) + lim
h!0
f (a + h) f (a)
h · h
= f (a) + f0(a)· 0
= f (a).
Remark 1:
(a) If f is continuous at x = a, it does not mean that f is differentiable at x = a.
(b) If f is not continuous at x = a, then f is not differentiable at x = a.
(c) If f is not differentiable at x = a, it does not mean that f is not continuous at x = a.
(d) A function f is not differentiable at x = a if one of the following is true:
i. f is not continuous at x = a.
ii. the graph of f has a vertical tangent line at x = a.
iii. the graph of f has a corner or cusp at x = a.
Teaching Tip
A lot of students erroneously deduce that the verb for “getting the derivative” is
“to derive”. Please correct this. The right verb is “to differentiate”. Moreover, the process of getting the derivative is “differentiation” — not “derivation”.
(C) EXERCISES:
1. Suppose f is a function such that f0(5)is undefined. Which of the following statements is always true?
a. f must be continuous at x = 5.
c. There is not enough information to determine whether or not f is continuous at x = 5. Answer: (a) False. Counterexample: any function that is not continuous at 5; (b) False. Counterexample f(x) = |x 5|; (c) True.
2. Which of the following statements is/are always true?
I. A function that is continuous at x = a is differentiable at x = a.
II. A function that is differentiable at x = a is continuous at x = a.
III. A function that is NOT continuous at x = a is NOT differentiable at x = a.
IV. A function that is NOT differentiable at x = a is NOT continuous at x = a.
a. none of them
Answer: only (e) is always true 3. Suppose that f is a function that is continuous at x = 3. Which of the following
statements are true?
a. f must be differentiable at x = 3. Answer: False, e.g. f(x) = |x + 3|.
b. f is definitely not differentiable at x = 3. Answer: False, e.g. f(x) = x.
4. Consider the function defined by
f (x) =
( x2 if x < 3, 6x 9 if x 3.
For each statement below, write True if the statement is correct and False, otherwise.
At x = 3, the function is
a. undefined. Answer: False
b. differentiable but not continuous. Answer: False
c. continuous but not differentiable. Answer: False
d. both continuous and differentiable. Answer: True
e. neither continuous nor differentiable. Answer: False
5. Determine the values of x for which the function is continuous.
a. f(x) = x + 5 6. Determine the largest subset of R where f(x) =p
25 x2 is continuous.
7. Is the function defined by g(x) = x2 sin x + 5 continuous at x = ⇡?
8. Is the function defined by f(x) = |x 1| differentiable at x = 1?
9. Is the function defined by
f (x) =
( x3 3 if x 2, x2+ 1 if x > 2.
continuous at x = 2? differentiable at x = 2?
10. Consider the function defined by f(x) =p
x. Is f differentiable at x = 1? at x = 0? at x = 1?