Fundamental increment lemma

In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative [math]\displaystyle{ f'(a) }[/math] of a function [math]\displaystyle{ f }[/math] at a point [math]\displaystyle{ a }[/math]:

[math]\displaystyle{ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}. }[/math]

The lemma asserts that the existence of this derivative implies the existence of a function [math]\displaystyle{ \varphi }[/math] such that

[math]\displaystyle{ \lim_{h \to 0} \varphi(h) = 0 \qquad \text{and} \qquad f(a+h) = f(a) + f'(a)h + \varphi(h)h }[/math]

for sufficiently small but non-zero [math]\displaystyle{ h }[/math]. For a proof, it suffices to define

[math]\displaystyle{ \varphi(h) = \frac{f(a+h) - f(a)}{h} - f'(a) }[/math]

and verify this [math]\displaystyle{ \varphi }[/math] meets the requirements.

The lemma says, at least when [math]\displaystyle{ h }[/math] is sufficiently close to zero, that the difference quotient

[math]\displaystyle{ \frac{f(a+h) - f(a)}{h} }[/math]

can be written as the derivative f' plus an error term [math]\displaystyle{ \varphi(h) }[/math] that vanishes at [math]\displaystyle{ h=0 }[/math].

I.e. one has,

[math]\displaystyle{ \frac{f(a+h) - f(a)}{h} = f'(a) + \varphi(h). }[/math]

Differentiability in higher dimensions

In that the existence of [math]\displaystyle{ \varphi }[/math] uniquely characterises the number [math]\displaystyle{ f'(a) }[/math], the fundamental increment lemma can be said to characterise the differentiability of single-variable functions. For this reason, a generalisation of the lemma can be used in the definition of differentiability in multivariable calculus. In particular, suppose f maps some subset of [math]\displaystyle{ \mathbb{R}^n }[/math] to [math]\displaystyle{ \mathbb{R} }[/math]. Then f is said to be differentiable at a if there is a linear function

[math]\displaystyle{ M: \mathbb{R}^n \to \mathbb{R} }[/math]

and a function

[math]\displaystyle{ \Phi: D \to \mathbb{R}, \qquad D \subseteq \mathbb{R}^n \smallsetminus \{ \mathbf{0} \}, }[/math]

such that

[math]\displaystyle{ \lim_{\mathbf{h} \to 0} \Phi(\mathbf{h}) = 0 \qquad \text{and} \qquad f(\mathbf{a}+\mathbf{h}) - f(\mathbf{a}) = M(\mathbf{h}) + \Phi(\mathbf{h}) \cdot \Vert\mathbf{h}\Vert }[/math]

for non-zero h sufficiently close to 0. In this case, M is the unique derivative (or total derivative, to distinguish from the directional and partial derivatives) of f at a. Notably, M is given by the Jacobian matrix of f evaluated at a.

We can write the above equation in terms of the partial derivatives [math]\displaystyle{ \frac{\partial f}{\partial x_i} }[/math] as

[math]\displaystyle{ f(\mathbf{a} + \mathbf{h}) - f(\mathbf{a}) = \displaystyle\sum_{i=1}^n \frac{\partial f(a)}{\partial x_i} + \Phi(\mathbf{h}) \cdot \Vert\mathbf{h}\Vert }[/math]

See also

• Generalizations of the derivative

References

• Talman, Louis (2007-09-12). "Differentiability for Multivariable Functions". http://clem.mscd.edu/~talmanl/PDFs/APCalculus/MultiVarDiff.pdf. Retrieved 2012-06-28. 
• Stewart, James (2008). Calculus (7th ed.). Cengage Learning. p. 942. ISBN 978-0538498845. 
• Folland, Gerald. "Derivatives and Linear Approximation". https://sites.math.washington.edu/~folland/Math134/lin-approx.pdf.