Mathematical Thought From Ancient to Modern Times: Volume 3, P1199 says
Another example of an impredicative definition is the definition of the maximum value of a continuous function defined over a closed interval as the largest value that the function takes on in this interval.
I cannot understand why the definition is impredicative, any one can give a clear explanation ?
When the definition is stated that way, we are essentially making a set $V$: the set of values that the function takes on the interval. Then we define the maximum value, $m$, as the largest value of $V$. But $m$ is already an element of $V$, which is why this can be seen as impredicative.
There are other definitions of the maximum value of a continuous function that are not impredicative. For example, $m$ is the maximum if and only if there is no rational $q$ in the interval so that $f(q) > m$, and for every positive rational $r$ there is a rational $q$ in the interval with $f(q) > m-r$. If we already have the rational numbers and the function $f$, then this definition implicitly characterizes $m$ - even if $m$ is not rational - but only quantifying over sets that we already have.