Page 11 - Pra U STPM 2022 Penggal 2 - Mathematics

Mathematics Semester 2 STPM Chapter 1 Limits and Continuity
Continuity on an interval

A function f is continuous on an open interval (a, b) if it is continuous at every number in the interval; a
function f is continuous on a closed interval [a, b] if it is continuous on (a, b) and is also continuous from the 1
right at a and from the left at b.
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Example 7

 2
Show that the function f defined by f(x) = 1 – x is continuous on the closed interval [–1, 1].

Solution: We first show that f is continuous on (–1, 1).
Let –1 , a , 1. Then
lim
lim
x → a f(x) = x → a 1 – x =  x ) = 1 – a = f(a)
 2
2
lim  2
(1 –
x → a
Next, we show that f is continuous from the right at –1 and from the left at x = 1.
lim
lim
+ (1 – x
lim  2
x → –1 + f(x) = x → –1 + 1 – x =  ) = 0 = f(–1)
2
x → –1
lim
lim
2
x → 1 – f(x) = x → 1 – 1 – x =  ) = 0 = f(1)
lim  2
– (1 – x
x → 1
Notice that a polynomial function is continuous on (–∞, ∞). A rational function is continuous on its domain.
x
The functions e , ln x, sin x, cos x, tan x, sin x, cos x and tan x are also continuous on their respective
–1
–1
–1
domains.
Exercise 1.2
1. The function f is defined by
sin x, x < 0,
f(x) =
x, x . 0.
π
Sketch the graph of f for – , x , π , and state whether f is continuous at 0.
2 2
2. The function f is defined by
x(x – 1), 0 < x , 2,
f(x) = 2(3 – x), 2 < x < 3.

Sketch the graph of f, and state whether it is continuous on its domain.
3. Sketch the graph of the function f in each of the following cases, and state whether f is a continuous
function.
(a) y = cos x, 0 < x < 2π (b) y = tan x, 0 < x < π
–x
2
(c) y = e , x  R (d) y = (x + 1) , x  R
(e) y = ln (1 + x), x . –1 (f) y = 2 , x  R, x ≠ 1
x – 1

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01 STPM Math(T) T2.indd 9 28/01/2022 5:30 PM