Four Flavors of Continuity

Here's a chart to help keep track of some of the different "flavors" of continuity in real analysis. Notice that the flavors vary according to $\delta$'s dependence on $\epsilon$, the point $x$, or the function $f$.
 
 
Explicitly, the definitions are give below. Let $X$ and $Y$ be metric spaces with metrics $d_X$ and $d_Y$, respectively.
  • Suppose $f:X\to Y$ is a function and fix $x_0\in X$. Then $f$ is continuous at $x_0$ if for each $x\in X$ and for each $\epsilon >0$ there is a $\delta>0$ such that $d_X(x,x_0)< \delta$ implies $d_Y(f(x),f(x_0))<\epsilon$.

  • Suppose $f:X\to Y$ is a function. Then $f$ is uniformly continuous if for each $\epsilon>0$ there is a $\delta>0$ such that $d_X(x_1,x_2)< \delta$ implies $d_Y(f(x_1),f(x_2))< \epsilon$ for all $x_1,x_2\in X$.

  • Let $\mathscr{F}$ be a collection of continuous functions $f:X\to Y$ and fix $x_0\in X$. Then $\mathscr{F}$ is equicontinuous at $x_0$ if for each $x\in X$ and for each $\epsilon>0$ there is a $\delta>0$ so that $d_X(x,x_0)< \delta$ implies $d_Y(f(x),f(x_0))<\epsilon$ for all $f\in\mathscr{F}$.

  • Let $\mathscr{F}$ be a collection of continuous functions $f:X\to Y$. Then $\mathscr{F}$ is uniformly equicontinuous if for each $\epsilon>0$ there is a $\delta>0$ so that $d_X(x_1,x_2)< \delta$ implies $d_Y(f(x_1),f(x_2))<\epsilon$ for all $x_1,x_2\in X$ and for all $f\in\mathscr{F}$.