Question

Judge the following is true or false If f is continuous on ( 0,1 ) then there is a c in (0,1) such that f(c) is a maximum value of f on (0,1)?

my friend tell me answer is true
but I think it is wrong
because according to the "Extreme Value Theorem"
the interval {a,b} should be "closed"
maximum is possible to be found
Is my thought wrong ?

Answer 1

False

Explanation:

As you believed, the interval would need to be closed for the statement to be true. To give an explicit counterexample, consider the function `f(x) = 1/x`.

`f` is continuous on `RR\\\\{0}`, and thus is continuous on `(0,1)`. However, as `lim_(x->0^+)f(x) = oo`, there is clearly no point `c in (0,1)` such that `f(c)` is maximal within `(0,1)`. Indeed, for any `c in (0,1)`, we have `f(c) < f(c/2)`. Thus the statement does not hold for `f`.