A [[Function]] or [[Function|Map]] $f$ is [[Continuous Function]] if all [[Limits|limits]] on $f$ [[Existential Quantifier|exist]]. $\huge \begin{align} f: X &\to Y \\ \forall a\in X,\exists y&: \lim_{ x \to a } f(a) = y \end{align} $ If a function is continuous on some [[Interval]] $[a,b]$, we denote it as $f \in \mathcal C[a,b]$. If there exists an $n$-th [[Derivative]] of $f$ on said [[Interval]] that is also continuous on the interval - we denote $f$ as $f\in \mathcal C^{n}$ Note the these sets of continuous functions and there derivatives are supersets of one another $\huge f \in \mathcal{C}^{k} \implies f\in \mathcal{C}^{k-1} \implies \cdots \implies f\in \mathcal{C}^{0} $