# Category:Definitions/Differentiable Real Functions

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This category contains definitions related to Differentiable Real Functions.

Related results can be found in Category:Differentiable Real Functions.

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:

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

exists.

#### Definition 2

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

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

exists.

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

## Subcategories

This category has only the following subcategory.

## Pages in category "Definitions/Differentiable Real Functions"

The following 20 pages are in this category, out of 20 total.

### C

### D

- Definition:Differentiable Mapping/Real Function
- Definition:Differentiable Mapping/Real Function/Interval
- Definition:Differentiable Mapping/Real Function/Interval/Closed Interval
- Definition:Differentiable Mapping/Real Function/Point
- Definition:Differentiable Mapping/Real Function/Point/Definition 1
- Definition:Differentiable Mapping/Real Function/Point/Definition 2
- Definition:Differentiable Mapping/Real Function/Real Number Line
- Definition:Differentiable on Interval
- Definition:Differentiable Real Function
- Definition:Differentiable Real Function at Point
- Definition:Differentiable Real Function in Open Set
- Definition:Differentiable Real Function on Closed Interval
- Definition:Differentiable Real Function on Open Interval