# 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 ?

As you believed, the interval would need to be closed for the statement to be true. To give an explicit counterexample, consider the function $f \left(x\right) = \frac{1}{x}$.
$f$ is continuous on $\mathbb{R} \setminus \setminus \setminus \setminus \left\{0\right\}$, and thus is continuous on $\left(0 , 1\right)$. However, as ${\lim}_{x \to {0}^{+}} f \left(x\right) = \infty$, there is clearly no point $c \in \left(0 , 1\right)$ such that $f \left(c\right)$ is maximal within $\left(0 , 1\right)$. Indeed, for any $c \in \left(0 , 1\right)$, we have $f \left(c\right) < f \left(\frac{c}{2}\right)$. Thus the statement does not hold for $f$.