Could someone please help me with this proof? I can't use the derivative

Ler f be a function defined on an interval I. We say That of is strictly increasing if x, ex , in I implies that f(xidef ( x2 ) Similarly, fis strictly decreasing if x, f(xx) Prove The following. f is continuous good injective on I, then of is strictly increasing on 2. If f is strictly increasing and if f(I) is an interval, Then f is continuous. Furthermore, Ah is a strictly increasing antitalons function on FRI). Can I use the Intermediate Value Theorem To prove This? Intermediare Value Theorem Suppose that f: [ab]-+ 1R is continuous. Then f has The intermediate value property on Cab]. That is, if Kis any value between flat and f(b) Cie., fla) ck