• The
sine function \sin: \mathbb R \rightarrow \mathbb R is bounded since |\sin (x)| \le 1 for all x \in \mathbb{R}. • The function f(x)=(x^2-1)^{-1}, defined for all real x except for −1 and 1, is unbounded. As
x approaches −1 or 1, the values of this function get larger in magnitude. This function can be made bounded if one restricts its domain to be, for example, [2, \infty) or (-\infty, -2]. • The function f(x)= (x^2+1)^{-1}, defined for all real
x,
is bounded, since |f(x)| \le 1 for all
x. • The
inverse trigonometric function arctangent defined as: y= \arctan (x) or x = \tan (y) is
increasing for all real numbers
x and bounded with -\frac{\pi}{2}
radians • By the
boundedness theorem, every
continuous function on a closed interval, such as f: [0, 1] \rightarrow \mathbb R, is bounded. More generally, any continuous function from a
compact space into a metric space is bounded. • All complex-valued functions f: \mathbb C \rightarrow \mathbb C which are
entire are either unbounded or constant as a consequence of
Liouville's theorem. In particular, the complex \sin: \mathbb C \rightarrow \mathbb C must be unbounded since it is entire. • The function f which takes the value 0 for x
rational number and 1 for
x irrational number (cf.
Dirichlet function)
is bounded. Thus, a function
does not need to be "nice" in order to be bounded. The set of all bounded functions defined on [0, 1] is much larger than the set of
continuous functions on that interval. Moreover, continuous functions need not be bounded; for example, the functions g:\mathbb{R}^2\to\mathbb{R} and h: (0, 1)^2\to\mathbb{R} defined by g(x, y) := x + y and h(x, y) := \frac{1}{x+y} are both continuous, but neither is bounded. (However, a continuous function must be bounded if its domain is both closed and bounded.) ==See also==