Could someone please help me with this proof and answer my question about countably many discontinuities

Let f be a function defined on an interval I. We say that f is strictly increasing if X, ex, in I implies that f(xidef(x) Similarly, fis strictly decreasing if x, 5x2 in I implies That f(x,) > f (xx) Prove The following. f is continuous and injective I , then of is secretly igerdesing on strictly decreasing 2. If f is strictly increasing and if f(I) is an interval, Then f is continuous. Furthercone, Ah is astrictly increasing continuous function of FRI). " Can I use the Intermediate Value Theoren To prove This? Intermediate Value Theorem Suppose That f: [ab]+ RR is continuous. Then f has The intermediate value property on Cab]. That is, if Kis any value between fla) and f(b) Cie., flack