Let $f\colon V\to W$ be a function, where $V$ and $W$ are Banach spaces. For $x\in V$, the function $f$ is said to be differentiable at $x$ if its derivative exists at that point. Differentiability at $x\in V$ implies continuity at $x$. If $S\subset V$, then $f$ is said to be differentiable on $S$ if $f$ is differentiable at every point $x\in S$.

For the most common example, a real function $f\colon\mathbb{R}\to\mathbb{R}$ is differentiable if its derivative $\frac{df}{dx}$ exists for every point in the region of interest. For another common case of a real function of $n$ variables $f(x_{1},x_{2},\ldots,x_{n})$ (more formally $f\colon\mathbb{R}^{n}\to\mathbb{R}$), it is not sufficient that the partial derivatives $\frac{\partial f}{\partial x_{i}}$ exist for $f$ to be differentiable. The derivative of $f$ must exist in the original sense at every point in the region of interest, where $\mathbb{R}^{n}$ is treated as a Banach space under the usual Euclidean vector norm.

If the derivative of $f$ is continuous, then $f$ is said to be $C^{1}$. If the $k$th derivative of $f$ is continuous, then $f$ is said to be $C^{k}$. By convention, if $f$ is only continuous but does not have a continuous derivative, then $f$ is said to be $C^{0}$. Note the inclusion property $C^{{k+1}}\subset C^{k}$. And if the $k$-th derivative of $f$ is continuous for all $k$, then $f$ is said to be $C^{\infty}$. In other words $C^{\infty}$ is the intersection $C^{\infty}=\bigcap_{{k=0}}^{\infty}C^{k}$.

Differentiable functions are often referred to as smooth. If $f$ is $C^{k}$, then $f$ is said to be $k$-smooth. Most often a function is called smooth (without qualifiers) if $f$ is $C^{\infty}$ or $C^{1}$, depending on the context.