Let $X$ and $Y$ be topological spaces. A function $f\colon X \to Y$ is continuous if, for every open set $U \subset Y$ the inverse image $f^{-1}(U)$ is an open subset of $X$

In the case where $X$ and $Y$ are metric spaces (e.g. Euclidean space, or the space of real numbers), a function $f\colon X \to Y$ is continuous if and only if for every $x \in X$ and every real number $\epsilon > 0$ there exists a real number $\delta > 0$ such that whenever a point $z \in X$ has distance less than $\delta$ to $x$ the point $f(z) \in Y$ has distance less than $\epsilon$ to $f(x)$

Continuity at a point

A related notion is that of local continuity, or continuity at a point (as opposed to the whole space $X$ at once). When $X$ and $Y$ are topological spaces, we say $f$ is continuous at a point $x \in X$ if, for every open subset $V \subset Y$ containing $f(x)$ there is an open subset $U \subset X$ containing $x$ whose image $f(U)$ is contained in $V$ Of course, the function $f\colon X \to Y$ is continuous in the first sense if and only if $f$ is continuous at every point $x \in X$ in the second sense (for students who haven't seen this before, proving it is a worthwhile exercise).

In the common case where $X$ and $Y$ are metric spaces (e.g., Euclidean spaces), a function $f$ is continuous at $x \in X$ if and only if for every real number $\epsilon > 0$ there exists a real number $\delta > 0$ satisfying the property that $d_Y(f(x),f(z)) < \epsilon$ for all $z \in X$ with $d_X(x,z) < \delta$ Alternatively, the function $f$ is continuous at $a \in X$ if and only if the limit of $f(x)$ as $x \to a$ satisfies the equation $$ \lim_{x \to a} f(x) = f(a). $$