A strictly incresing or decresing function that the Domain and Image are intervals is a continuous function?

Question

How can I begin to prove this: Let f be a function such that the Domain and Image are intervals. Prove that if f is strictly increasing (or strictly decreasing), then f is continuous.

Answer 1

Suppose that $f$ is strictly increasing and that it is discontinuous at some point $a$. Then $\lim_{x\to a}f(x)\neq f(a)$. Since $f$ is strictly increasing, $\lim_{x\to a^-}f(x)\leqslant f(a)\leqslant\lim_{x\to a^+}f(x)$, and therefore you must have $\lim_{x\to a^-}f(x)<f(a)$ or $f(a)<\lim_{x\to a^+}f(x)$. In the first case, the interval $\left(\lim_{x\to a^-}f(x),f(a)\right)$ is outside the range of $f$ and in the second case the interval $\left(a,\lim_{x\to a^+}f(x)\right)$ is outside the range of $f$. In any case (and now the fact that $f$ is strictly increasing is needed once again), the range of $f$ cannot be an interval.

Answer 2

Let $f$ be strictly increasing. Then its discontinuities (if any) are jump discontinuities. Suppose $f$ jumps from a value $a$ to a value $b >a$ when you cross $c$. Then no number between $a$ and $b$ is in the range of $f$. Since the range is assumed to be an interval This cannot happen. Hence $f$ has no discontinuity points.