$C^0$: 连续函数。

$C^k$: 函数是连续的，且其 $k$ 阶导数连续。

$C^{\infty }$: 光滑函数。

$L_1(\Omega )$: $\Omega$ 上Lebesgue-可积函数全体。

$L_{\infty }(\Omega )$: $\Omega$ 上有界函数全体。

Consider an open set $U$ on the real line and a function $f$ defined on $U$ with real values. Let $k$ be a non-negative integer. The function $f$ is said to be of differentiability class $C^k$ if the derivatives $f^{\prime },f^{″},\dots ,f^{(k)}$ exist and are continuous on $U$. If $f$ is $k$-differentiable on $U$, then it is at least in the class $C^{k-1}$ since $f^{\prime },f^{″},\dots ,f^{(k-1)}$ are continuous on $U$. The function $f$ is said to be infinitely differentiable, smooth, or of class $C^{\infty }$, if it has derivatives of all orders on $U$. (So all these derivatives are continuous functions over $U$.)

若一函数是连续的，则称其为 $C^0$ 函数；若函数 $1$ 阶可导，且其 $1$ 阶导函数连续，则被称为 $C^1$ 函数；若 $n$ 阶可导，且其 $n$ 阶导函数连续，则为 $C^n$ 函数。而光滑函数是任意阶导函数都存在且连续的函数，记为 $C^{\infty }$。

参考文献：
https://en.wikipedia.org/wiki/Smoothness