Could someone please check my proof

Ler f be a function defined on an interval I. Prove That if of is strictly increasing and if f (I) is an interval, Then of is continuous. Furthermore, f" is a strictly increasing Continuous function on f(I). Ler I = Lab]. Since f is strictly increasing Then for all Xne Xay in I There is a Andall K between F(x ) and f (x,.. ); Sot is always possible To find a point CE ( Xn, Xny) where f(c ) = K. Since all f(x,)