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Homedifferentiable function

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# 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 $f\colon\mathbb{R}\to\mathbb{R}$ 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.

## Mathematics Subject Classification

26A24*no label found*57R35

*no label found*

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## Comments

## functions on a fin. dim. vector space

In regards to the point raised in correction number 4.

Let's say V is an n-dimensional, real vector space.

1) Note that V is not a banach space, although it could be made

into one in an infinite number of ways.

2) It is true that V is isomorphic to R^n, in the sense

that there exist linear bijections between V and R^n,

however there is no way to prefer any one such

bijection over another.

We would like to define what it means for a

function f:V -> R to be differentiable, but how

to proceed?

We need a norm for the denominator of our limit

expression, but which norm are we to use?

The fact of the matter is that if f:V->R is differentiable

with respect to one norm, it is differentiable with

respect to all norms. This is an interesting consequence

of the finite-dimensionality of V.

Thus the concept of "differentiable function"

makes sense in the context of "finite dimensional

vector spaces", which is not quite the same context as

"Banach spaces".

A similar phenomenon occurs if we try to topologize

V. The way to proceed is to pick a norm, any norm

and to use that particular norm topology. It doesn't

matter which norm we choose to do this, we get the

same topology regardless. This is, of course,

not true in the infinite-dimensional

setting.

## See also

non-Newtonian calculus