Problem
Let $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is continuous and injective. Prove (or disprove) that $f$ is a strictly monotone function.
I have tried to prove the monotonocity of $f$ for then by the injectivity of $f$ strict monotonicity will follow. However, I couldn't make any success.
I also tried to show that,
Let $f:\mathbb{R}\to\mathbb{R}$ such that $f$ is not a strictly monotone function. Then show that $f$ is either discontinuous or not injective.
In this case also I couldn't proceed beyond some trivial observations.
Can anyone help me to prove it?