differentiable function DifferntiableFunction 2002-05-19 03:20:51 2006-06-08 19:06:39 Definition differentiable smooth differentiable smooth % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{mathrsfs} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newcommand{\C}{\mathbb{C}} \newcommand{\R}{\mathbb{R}} \newcommand{\N}{\mathbb{N}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Per}{\operatorname{Per}} 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 \emph{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 {\em 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.