Let $g : [ 0,1 ] \rightarrow [ 0,1 ]$ be an increasing function (but not necessarily continuous). Show that $g$ has at least one fixed point. Hint: one may consider the set $$E = \{ x \in [ 0,1 ] ; x \leqslant g ( x ) \} .$$
Let $g : [ 0,1 ] \rightarrow [ 0,1 ]$ be an increasing function (but not necessarily continuous). Show that $g$ has at least one fixed point. Hint: one may consider the set
$$E = \{ x \in [ 0,1 ] ; x \leqslant g ( x ) \} .$$