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 known as
Some sources refer to this as the continuity property, but this is not standard, and is too easily confused with the Continuum Property.

Also see

 * Extreme Value Theorem