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# Question
What is a continuous function?
# Answer
A [[Continuous|continuous]] function is a function that has does not abruptly change in value.
We say that $f$ is continuous (at every $x$) if $f^{-1}(A) \equiv \{ x \in X \ |\ f(x) \in A \}$ is [[Open Sets|open]] for every [[Open Sets|open]] $A \subset Y$.
> [!example] Example of a Non-Continuous
> $f(x) = \frac{1}{x}$ is not a [[Continuous|continuous]] function as at $x=0$ there is an abrupt change in value, and the value there is undefined.

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# Question
Why does a real valued continuous function obtain its maximum on a compact set?
# Answer