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, $$J$$ 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.

Note
Not to be confused with the Continuum Property.