Web Reference: Dec 14, 2025 · Both discrete and continuous variables generally do have changing values—and a discrete variable can vary continuously with time. I am quite aware that discrete variables are those values that you can count while continuous variables are those that you can measure such as weight or height. A function is continuous if the preimage of every open set is an open set. (This is the definition in topology and is the "right" definition in some sense.) The definitions you cite of semicontinuities claim that the preimages of certain open sets are open, but does not say so about all open sets. Note that $\ { \ {f \in \mathbb {R} \mid f > \alpha\} \mid \alpha \in \mathbb {R} \} \cup ... Jan 5, 2016 · As such, $\arctan$ is continuous. If you define $\arctan$ by integrals or power series the result is immediate (the first by the Lipshitz continuity of the indefinite integral and the second from the uniform convergence of power series in compact sets)
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