Page 131 - DMTH401_REAL ANALYSIS

Unit 9: Functions

9.4.7 Bounded Functions Notes

In Unit 2, you were introduced to the notion of a bounded set, upper and lower bounds of a set.
Let us now extend these notions to a function.

You know that if f : S  R is a function, (S C R), then
{f(x) : x S)}

is called the range set or simply the range of the function f.
A function is said to be bounded if its range is bounded.
Let f : SR be a function. It is said to be bounded above if there exists a real number K such that

f(x)  k " x S
The number K is called an upper bound of it. The function f is said to be bounded below if there
exists a number k such that
f(x)  k " x S

The number k (is called a lower bound of f).
A function f : S R, which is bounded above as well as bounded below, is said to be bounded.
This implies that there exist two real numbers k and K such that
k  f(x)  K " x S.
This is equivalent to say that a function f : S R is bounded if there exists a real number M such
that
|f(x)|  M, " x S.
A function may be bounded above only or may be bounded below only or neither bounded
above nor bounded below.
Recall that sin x and cos x are both bounded functions. Can you say why? It is because of the
reason that the range of each of these functions is [–1, 1].

Example: A function f : R  R defined by
2
(i) f(x) = –x , " x R is bounded above with 0 as an upper bound
(ii) f(x) = x, " x  0 is bounded below with 0 as a lower bound

(iii) f(x) = for |x| –l is bounded because |f(x)|  1 for |x||.

Self Assessment

1. Test whether the following are rational numbers:
(i) I7 (ii) 8 (iii) 3 + 2

2
2. The inequality x – 5x + 6 < 0 holds for
(i) x < 2, x < 3 (ii) x > 2, x < 3
(iii) x < 2, x > 3 (iv) x > 2, x > 3

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