What is the definition of continuity?

When I first studied real analysis, the definition of continuity was clear but there were just so many of them. I took some time to list as many definition of continuity as I could, and provide some observations on why we need so many definitions and how they are different.

Topological Definition of Continuity

The topological definition of continuity is as follows:

Open set Continuity

(Open set continuity) Given a topological space $X$ and $Y$, a function $f : X \to Y$ is continuous if for every open set $\mathcal{O} \in Y$ the inverse image $f^{-1}(\mathcal{O}) = { x \in X : f(x) \in \mathcal{O}}$ is an open subset of $X$

This rather general definition of continuity can be quite baffling. There’s another topological definition of continuity:

Sequential Continuity

(Sequential continuity) Given a topological space $X$ and $Y$, a function $f : X \to Y$ is sequentially continuous $f(x_n) \to f(x)$ whenever $x_n \to x$. (Remark: Note that sequential continuity does NOT imply continuity but continuity does imply sequential continuity.)

However, in a metric space, the above two definitions are equivalent.

Metric space Definition of Continuity

One can easily see that metric space definitions require a distance function (obviously) instead of open sets or sequences. Here we introduce some of the most fundamental definitions of continuity:

Point-wise continuity

(Point-wise continuity) Given a metric space $(X, d_1)$ and $(Y, d_2)$, a function $f: X \to Y$ is continuous at point $x \in X$ if for every $\epsilon > 0$ there exists $\delta > 0$ such that $d_2(f(x), f(y)) < \epsilon$ whenever $d_1(x,y) < \delta$ (Remark: note that point-wise continuity is clearly a local property).

Uniform continuity

(Uniform continuity) Given a metric space $(X, d_1)$ and $(Y, d_2)$, a function $f: X \to Y$ is uniformly continuous on $X$ if for every $\epsilon > 0$ there exists a $\delta > 0$ such that whenever $d_1(x,y) < \delta$, we have $d_2(f(x), f(y)) < \epsilon$ (Remark: note that uniform continuity is a local property).

As opposed to the topological definition, the $\delta\text{-}\epsilon$ continuity definitions require a metric space.

There’s even more definition of continuity:

Lipschitz continuity

(Lipschitz continuity) Given two metric spaces $(X, d_X)$ and $(Y, d_Y)$, a function $f: X \to Y$ is Lipschitz continuous if there exists a real constant $K \ge 0$ such that for all $x_1$ and $x_2$ in $X$,

\[d_Y(f(x_1), f(x_2)) \le Kd_X(x_1, x_2)\]

Any such $K$ is called a Lipschitz constant for the function $f$. (Remark: note that Lipschitz continuity is a global property) .

a-Hölder Condition

($\alpha$-Holder condition) Given two metric spaces $(X, d_X)$ and $(Y, d_Y)$, a function $f: X \to Y$ satisfies $\alpha$-Holder condition (or $\alpha$-Holder continuous) if there exists $C_\alpha$ such that

\[d_Y(f(x), f(x')) \le C_\alpha d_X(x, x') \; \forall x,x' \in X\]

(Remark: note that Holder condition is a global property) .

The above two definitions also rely on metric spaces. However, unlike $\delta\text{-}\epsilon$ definitions which are local properties, the above continuities are global properties, hence they are a stronger.

Why so many definitions?

Continuity with different degree of constraints

Clearly there are lots of definitions of continuity but there is a good reason for it. Some continuity conditions are stronger than others. For instance, uniform continuity is a defines continuity on a set as opposed to point-wise continuity that defines it on a single point. Similarly, Lipschitz continuity and holder condition are both global property with very strict condition that all pairs satisfy the bound.

Why topological definition of continuity?

Then why do we have a topological definition and a metric space definition? Well the topological space is more general than the metric space (metric spaces are topological spaces with additional constraints). Hence the definition of continuity on a topological space also hold in metric spaces.