# differentiable function

Let $f:V\to W$ 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 $f:\mathbb{R}\to \mathbb{R}$ 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
$f({x}_{1},{x}_{2},\mathrm{\dots},{x}_{n})$ (more formally $f:{\mathbb{R}}^{n}\to \mathbb{R}$),
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 ${C}^{\mathrm{\infty}}$. In other words ${C}^{\mathrm{\infty}}$ is the
intersection^{} ${C}^{\mathrm{\infty}}={\bigcap}_{k=0}^{\mathrm{\infty}}{C}^{k}$.

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 ${C}^{\mathrm{\infty}}$ or ${C}^{1}$, depending on the context.

Title | differentiable function |

Canonical name | DifferentiableFunction |

Date of creation | 2013-03-22 12:39:10 |

Last modified on | 2013-03-22 12:39:10 |

Owner | Koro (127) |

Last modified by | Koro (127) |

Numerical id | 24 |

Author | Koro (127) |

Entry type | Definition |

Classification | msc 26A24 |

Classification | msc 57R35 |

Synonym | smooth function |

Synonym | differentiable mapping |

Synonym | differentiable map |

Synonym | smooth mapping |

Synonym | smooth map |

Synonym | continuously differentiable |

Related topic | OneSidedDerivatives |

Related topic | RoundFunction |

Related topic | ConverseTheorem |

Related topic | WeierstrassFunction |

Defines | differentiable |

Defines | smooth |