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Decreasing Function


A function f(x) decreases on an interval I if f(b)<=f(a) for all b>a, where a,b in I. If f(b)<f(a) for all b>a, the function is said to be strictly decreasing.

Conversely, a function f(x) increases on an interval I if f(b)>=f(a) for all b>a with a,b in I. If f(b)>f(a) for all b>a, the function is said to be strictly increasing.

If the derivative f^'(x) of a continuous function f(x) satisfies f^'(x)<0 on an open interval (a,b), then f(x) is decreasing on (a,b). However, a function may decrease on an interval without having a derivative defined at all points. For example, the function -x^(1/3) is decreasing everywhere, including the origin x=0, despite the fact that the derivative is not defined at that point.


See also

Derivative, Increasing Function, Nondecreasing Function, Nonincreasing Function, Strictly Decreasing Function, Strictly Increasing Function

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References

Jeffreys, H. and Jeffreys, B. S. "Increasing and Decreasing Functions." §1.065 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 22, 1988.

Referenced on Wolfram|Alpha

Decreasing Function

Cite this as:

Weisstein, Eric W. "Decreasing Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DecreasingFunction.html

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