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Calculus 1.4 Continuity and One-Sided Limits (Part b)
Yesterday we looked at continuity on an open interval so today it is only natural
that we look at continuity on a closed interval. To do this we first have to talk
about a one-sided limit. One sided limits happen when we look at what is
happening to a function, f, on only the right or left side if a member of it’s
domain, c.
Limit from the left:
Limit from the right:
lim f ( x)
(we look at x’s close to c but less than c)
x → c−
lim f ( x)
(we look at x’s close to c but greater than c)
x → c+
Ex: Find the limit of the following
lim
2[x] − 1
lim
2[x ] − 1
x → 5+
lim
x → 5−
x2 − 3
x → 5+
lim
x2 − 3
x → 5−
THEOREM 1.10 The Existence of a Limit
Let f be a function and le t c and L be real numbers. The limit of f(x) as x
approaches c is L if and only if
lim f ( x)
x→c
−
=
L
=
lim f ( x)
x → c+
A function is continuous on a closed interval if it is continuous in the interior and
possesses one-sided continuity at the endpoints.
Ex: Identify an interval for which the following functions are continuous.
(a) f ( x) = − x
(b) f ( x) = 4 − x
(c ) f ( x) = 9 − x 2
THEOREM 1.11 Properties of Continuity
If b is a real number and f and g are continuous at x = c, then the following
functions are also continuous at c.
1. Scalar multiple: bf
2. Sum and difference: f ± g
3. Product: fg
4. Quotient:
f
, if
g
g (c) ≠ 0
$$ The following types of functions are continuous at every point in their
domains: $$
1. Polynomial Functions: an x n + an −1 x n −1 + an −2 x n −2 + ... + a2 x 2 + a1 x + a 0
2. Rational Functions: r ( x) =
p ( x)
,
q( x)
q (c) ≠ 0
3. Radical Functions: f ( x) = n x
4. Trigonometric Functions: sinx, cosx, tanx, cotx, secx, cscx
Ex: Describe the interval(s) on which each function is continuous.
 x−2 
(b) f ( x) = ( x + 1) 2

 x − x
(a) f ( x) = cot x
(c) f ( x) =
4
x −1
THEOREM 1.13 Intermediate Value Theorem:
If f is continuous on the closed interval [a, b] and k is any number between f(a)
and f(b), then there is at least one number c in [a, b], such that f(c) = k.
Ex: Verify that the Intermediate Value Theorem applies to the indicated
interval and find the value of c guaranteed by the theorem.
f ( x) = x 2 − 6 x + 8,
[0, 3],
f (c ) = 0
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