Definition:Differentiable Mapping/Real Function

Definition

At a Point

Let $f$ be a real function defined on an open interval $\openint a b$.

Let $\xi$ be a point in $\openint a b$.

Definition 1

$f$ is differentiable at the point $\xi$ if and only if the limit:

$\ds \lim_{x \mathop \to \xi} \frac {\map f x - \map f \xi} {x - \xi}$

exists.

Definition 2

$f$ is differentiable at the point $\xi$ if and only if the limit:

$\ds \lim_{h \mathop \to 0} \frac {\map f {\xi + h} - \map f \xi} h$

exists.

These limits, if they exist, are called the derivative of $f$ at $\xi$.

On an Open Interval

Let $f$ be a real function defined on an open interval $\openint a b$.

Then $f$ is differentiable on $\openint a b$ if and only if $f$ is differentiable at each point of $\openint a b$.

On the Real Number Line

Let $f$ be a real function defined on $\R$.

By definition, $\R$ is an (unbounded) open interval.

Let $f$ be differentiable on the open interval $\R$.

That is, let $f$ be differentiable at every point of $\R$.

Then $f$ is differentiable everywhere (on $\R$).

Also see

• Definition:Continuously Differentiable Real Function
• Definition:Differentiability Class of Real Function

• Results about differentiable real functions can be found here.

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 10.1$
• 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards: Calculus (8th ed.): $\S 2.1$