Continuous Image of Closed Interval is Closed Interval

Theorem
Let $f$ be a real function which is continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.

Then the image of $\left[{a \,.\,.\, b}\right]$ under $f$ is also a closed interval.

Proof
Let $I = \left[{a \,.\,.\, b}\right]$.

Let $J = f \left({I}\right)$.

From Image of Interval by Continuous Function is Interval, $J$ is an interval.

From Image of Closed Real Interval is Bounded, $J$ is bounded.

From Max and Min of Function on Closed Real Interval‎, $J$ includes its end points.

Hence the result.

Also see
Not to be confused with the Continuum Property.