Find the maximum and the minimum values,if any,of the function given by $f(x) = x, x \in (0, 1)$.

(NONE) The given function $f(x) = x$ is a strictly increasing function in the open interval $(0, 1)$.
From the graph of the function,it appears that the minimum value should be at a point closest to $0$ on its right,and the maximum value should be at a point closest to $1$ on its left.
However,such points do not exist in the open interval $(0, 1)$.
For any point $x_0 \in (0, 1)$,we can always find a smaller point $\frac{x_0}{2} \in (0, 1)$ such that $\frac{x_0}{2} < x_0$. Thus,there is no minimum value.
Similarly,for any point $x_1 \in (0, 1)$,we can always find a larger point $\frac{x_1 + 1}{2} \in (0, 1)$ such that $\frac{x_1 + 1}{2} > x_1$. Thus,there is no maximum value.
Therefore,the function $f(x) = x$ has neither a maximum nor a minimum value in the interval $(0, 1)$.