Monotonic function - Wikipedia, the free encyclopedia
See "Some basic applications and results".
Consider a function f: R-------R
Suppose f is increasing. Is it necessary that that f must approach to infinity?
For example if f=1/x then it is decreasing and approaches to 0 in (0, infinity) .but that that is possible only when domain is restricted. Is it possible on R for some function which are either increasing or decreasing on R but does not approach to +infinity or -infinity?
We can take 1/x when x >= 1, reflect it with respect to x-axis to get -1/x, shift it left by 1 and up by 1 to get g(x) = -1/(x + 1) + 1. Here is its graph plotted by WolframAlpha. Then define . I.e., to get the graph of f(x) for x < 0, we rotate the graph of g(x) for x >= 0 by 180 degrees. The result is a smooth function that tends to -1 when and to 1 when .
For another example, consider arctan(x).