The range of a continuous function

Theorem 7846

If $f:[a,b]\rightarrow (-\infty,\infty)$ is a continuous function then the range of f is a closed interval.

For example, we have that the range of $f:[0,2]\rightarrow(-\infty,\infty)$given by f(x) = x² is [0,4] and the range of $f:[0,4\pi]\rightarrow(-\infty,\infty)$ given by $f(x) = \sin(x)$ is [-1,1].

It follows from this theorem that if the domain of a continuous function is any sort of interval, and for some x, y and c we have f(x) < c < f(y) then for some z we have f(z) = c. This is the intermediate value property, and is what allows us to conclude, for example, that the equation $e^x + \ln(x) - 4 = 0$ has a solution.

It also follows from this theorem that if the domain of a continuous function is a closed interval then there are numbers x and y so that for all z in the domain of f, $f(x) \leq f(z) \leq f(y)$. The number f(y) is called the maximum value of f, and the number f(x) is called the minimum value of f. The maximum and minimum values of such a function are nothing more than the right and left endpoints of its range. Thus the theorem allows us to conclude that a continuous function whose domain is a closed interval achieves its maximum and minimum values.

The theorem gives us no information about discontinuous function nor direct information about the range of a continuous function whose domain is not a closed interval. The intermediate value property does allow us to conclude that the range of a continuous function whose domain is an interval is also an interval. For example, if $f:(0,\infty)\rightarrow(0,\infty)$ is given by f(x) + x + x^-1 then the range of f is $[2,\infty)$ as can be seen by observing three things:

• f(x) = (x^1/2 - x^-1/2)² + 2, so the range of f is a subset of $[2,\infty)$ and includes 2 = f(1);
• The range of f is an interval, since f is continuous and its domain is an interval;
• The range of f is not contained in any interval of finite length, since $\lim_{x\rightarrow \infty} f(x) = +\infty$.