Properties of Continuous Functions on Closed Intervals

Let the function $f(x)$ be defined on the closed interval $[a, b]$. If:

1. $f(x)$ is continuous on the open interval $(a, b)$
2. $f(x)$ is right-continuous at $a$: $\lim_{x \to a^+} f(x) = f(a)$
3. $f(x)$ is left-continuous at $b$: $\lim_{x \to b^-} f(x) = f(b)$

then $f(x)$ is called continuous on the closed interval $[a, b]$.