##  [Differentiability](/differentiability-0) 

 Definition

Property that a function admits a linear approximation at a point: there exists a linear map (the derivative) that approximates first-order increments so that f(x0+h) = f(x0) + Df(x0)[h] + o(‖h‖). In R this reduces to the limit of the difference quotient.

 

 

 

 

 

 





## Principle

Principle

First-order linearization: the leading behaviour of the increment is captured by a linear map, and the remainder is of strictly smaller order than the input increment.

 

 

 

 

 





## Demonstration

Demonstration

For f(x)=x^2 on R, the derivative at x0 is Df(x0)=2x0 because f(x0+h)-f(x0)=2x0 h + h^2, and the h^2 term is o(h). The absolute value |x| is not differentiable at 0 because the one-sided difference quotients disagree.

 

 

 

 

## Misapplication

Misapplication

Assuming differentiability is equivalent to continuity (differentiability implies continuity but not conversely), or assuming differentiability everywhere from information at a dense set of points—pointwise differentiability need not extend to a neighbourhood.

 

 

 

 

 





## Consequence

Consequence

Differentiability yields linear approximations, the chain rule, and the mean value theorem under mild hypotheses; higher regularity (C^k, smoothness) follows by existence of successive derivatives. In finite dimensions differentiability implies local approximability by tangent maps.

 

 

 

 

## Reversal

Reversal

Non-differentiability occurs where no linear map gives the required first-order approximation despite possible continuity. Examples include corners, cusps, and certain fractal graphs which are continuous but fail to be differentiable.

 

 

 

 

 





## Boundary

Boundary

Differentiate notions: Fréchet differentiability in Banach spaces, Gâteaux differentiability as weaker directional derivative concept, and classical single-variable derivative. Differentiability requires linear structure and normed setting for the standard definition.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension with smoothness and analyticity: differentiability (existence of a derivative) does not automatically imply smoothness (existence of all higher derivatives) nor analyticity (power series expansion); these are strictly stronger notions.

 

 

 

 

 





## Synthesis

Synthesis

Differentiability is the requirement that a function has a best linear first-order approximation at a point; it upgrades continuity to a quantitative local linear model and is the basis for calculus rules and linearization techniques.