The Intermediate Value Theorem (IVT) says functions that are continuous on an interval [a,b] take on all (intermediate) values between their extremes. The Extreme Value Theorem (EVT) says functions that are continuous on [a,b] attain their extreme values (high and low).

Here’s a statement of the EVT: Let f be continuous on [a,b]. Then there exist numbers c,din [a,b] such that f(c)leq f(x)leq f(d) for all xin [a,b]. Stated another way, the “supremum ” M and “infimum ” m of the range {f(x):xin [a,b]} exist (they’re finite) and there exist numbers c,din [a,b] such that f(c)=m and f(d)=M.

Note that the function f must be continuous on [a,b] for the conclusion to hold. For example, if f is a function such that f(0)=0.5, f(x)=x for 0