differentiable function

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 $f\colon\R\to\R$ 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 $f\colon\R^n\to\R$ ), 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 $\R^n$ 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.