If is strictly monotonic it means if , say WLOG then or . Thus, we see . This implies that is one-to-one.

Now we will show that if is one-to-one then is strictly monotonic. This part is more involved. The hint that will help you solve this is if: then lies between . We will prove the hint and you can try finishing the proof. If did not lie between it means . Pick a real number so that . Then by intermediate value theorem there is . But then for . A contradiction.