What is the meaning of bounded function in general term. What happens when we put a boundary condition on a function.

Question

What is the meaning of bounded function in general term? What happens when we put a boundary condition on a function.

Answer 1

A bounded function is a function that its range can be included in a closed interval. That is for some real numbers $a$ and $b$ you get $a\le f(x) \le b$ for all $x$ in the domain of $f$.

For example $f(x)= \sin x $ is bounded because for all values of $x$, $-1\le \sin x \le 1 $.

Note that boundedness depends on the domain of the function.

For example, $f(x)=x^2$ is bounded on $[-10, 10]$ but it is not bunded on$(- \infty, \infty)$

Answer 2

In general a function f:X -> Y is is bounded when:
Y is a topological space and f(X) is subset of a compact set;
Y is a metric space and diam f(X) is finite;
Y is a poset and f(X) has a lower and upper bound.

A boundary condition is a condition placed upon a collection
of functions such as the solutions of a differential equation.

If y' = 5, then y(x) = 5x + c for some c. Requiring
y(0) = 25 is a boundary condition making y(x) = 5x + 25.