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Let $(X, \leq_{X})$, $(Y, \leq_{Y})$ be Total Orderings equipped with their respective Order Topologys and let $f: X \to Y$ be a continuous Non-Increasing Function. Let $A \subset X$ and suppose $\sup\limits A$ exists. Then $$f(\sup\limits A) = \inf\limits f(A)$$

By definition of Infimum, $\inf\limits_{\leq_{Y}} f(A) = \sup\limits_{\leq_{Y}’} f(A)$, where $\leq_{Y}’$ is the Reverse Ordering on $Y$. Since $f$ is a continuous Non-Decreasing Function to the Reverse Ordering, Non-Decreasing Continuous Functions preserve Supremum tells us that $$f(\sup\limits A) = \sup\limits_{\leq_{Y}’}f(A) = \inf\limits_{\leq_{Y}}f(A).$$ $\blacksquare$