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1.7. LIMITS, CONTINUITY, AND DIFFERENTIABILITY
More formally, we make the following definition.
Definition 1.7. A function f is continuous at x = a provided that
(a) f has a limit as x → a,
(b) f is defined at x = a, and
(c) lim
x→a
f (x) = f (a).
Conditions (a) and (b) are technically contained implicitly in (c), but we state them
explicitly to emphasize their individual importance. In words, (c) essentially says that a
function is continuous at x = a provided that its limit as x → a exists and equals its
function value at x = a. If a function is continuous at every point in an interval [a, b], we
say the function is “continuous on [a, b].” If a function is continuous at every point in
its domain, we simply say the function is “continuous.” Thus, continuous functions are
particularly nice: to evaluate the limit of a continuous function at a point, all we need to
do is evaluate the function.
For example, consider p(x) = x 2 − 2x + 3. It can be proved that every polynomial is a
continuous function at every real number, and thus if we would like to know lim x→2 p(x),
we simply compute
lim
x→2
(x
2 − 2x + 3) = 2
2 − 2 · 2 + 3 = 3.
This route of substituting an input value to evaluate a limit works anytime we know the
function being considered is continuous. Besides polynomial functions, all exponential
functions and the sine and cosine functions are continuous at every point, as are many
other familiar functions and combinations thereof.
Activity 1.19.
This activity builds on your work in Preview Activity 1.7, using the same function f as
given by the graph that is repeated in Figure 1.39
(a) At which values of a does lim x→a f (x) not exist?
(b) At which values of a is f (a) not defined?
(c) At which values of a does f have a limit, but lim x→a f (x) f (a)?
(d) State all values of a for which f is not continuous at x = a.
(e) Which condition is stronger, and hence implies the other: f has a limit at x = a
or f is continuous at x = a? Explain, and hence complete the following sentence: “If f
at x = a, then f
at x = a,” where you complete the blanks with has a limit and is continuous,
using each phrase once.
⊳
1.7. LIMITS, CONTINUITY, AND DIFFERENTIABILITY
More formally, we make the following definition.
Definition 1.7. A function f is continuous at x = a provided that
(a) f has a limit as x → a,
(b) f is defined at x = a, and
(c) lim
x→a
f (x) = f (a).
Conditions (a) and (b) are technically contained implicitly in (c), but we state them
explicitly to emphasize their individual importance. In words, (c) essentially says that a
function is continuous at x = a provided that its limit as x → a exists and equals its
function value at x = a. If a function is continuous at every point in an interval [a, b], we
say the function is “continuous on [a, b].” If a function is continuous at every point in
its domain, we simply say the function is “continuous.” Thus, continuous functions are
particularly nice: to evaluate the limit of a continuous function at a point, all we need to
do is evaluate the function.
For example, consider p(x) = x 2 − 2x + 3. It can be proved that every polynomial is a
continuous function at every real number, and thus if we would like to know lim x→2 p(x),
we simply compute
lim
x→2
(x
2 − 2x + 3) = 2
2 − 2 · 2 + 3 = 3.
This route of substituting an input value to evaluate a limit works anytime we know the
function being considered is continuous. Besides polynomial functions, all exponential
functions and the sine and cosine functions are continuous at every point, as are many
other familiar functions and combinations thereof.
Activity 1.19.
This activity builds on your work in Preview Activity 1.7, using the same function f as
given by the graph that is repeated in Figure 1.39
(a) At which values of a does lim x→a f (x) not exist?
(b) At which values of a is f (a) not defined?
(c) At which values of a does f have a limit, but lim x→a f (x) f (a)?
(d) State all values of a for which f is not continuous at x = a.
(e) Which condition is stronger, and hence implies the other: f has a limit at x = a
or f is continuous at x = a? Explain, and hence complete the following sentence: “If f
at x = a, then f
at x = a,” where you complete the blanks with has a limit and is continuous,
using each phrase once.
⊳