Fundamental increment lemma

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

f ′ ( a ) = lim h → 0 f ( a + h ) − f ( a ) h . {\displaystyle f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}.}

The lemma asserts that the existence of this derivative implies the existence of a function φ {\displaystyle \varphi } such that

lim h → 0 φ ( h ) = 0 and f ( a + h ) = f ( a ) + f ′ ( a ) h + φ ( h ) h {\displaystyle \lim _{h\to 0}\varphi (h)=0\qquad {\text{and}}\qquad f(a+h)=f(a)+f'(a)h+\varphi (h)h}

for sufficiently small but non-zero h {\textstyle h}. For a proof, it suffices to define

φ ( h ) = f ( a + h ) − f ( a ) h − f ′ ( a ) {\displaystyle \varphi (h)={\frac {f(a+h)-f(a)}{h}}-f'(a)}

and verify this φ {\displaystyle \varphi } meets the requirements.

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

f ( a + h ) − f ( a ) h {\displaystyle {\frac {f(a+h)-f(a)}{h}}}

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

That is, one has

f ( a + h ) − f ( a ) h = f ′ ( a ) + φ ( h ) . {\displaystyle {\frac {f(a+h)-f(a)}{h}}=f'(a)+\varphi (h).}

Differentiability in higher dimensions

In that the existence of φ {\displaystyle \varphi } uniquely characterises the number f ′ ( a ) {\displaystyle f'(a)}, 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 R n {\displaystyle \mathbb {R} ^{n}} to R {\displaystyle \mathbb {R} }. Then f is said to be differentiable at a if there is a linear function

M : R n → R {\displaystyle M:\mathbb {R} ^{n}\to \mathbb {R} }

and a function

Φ : D → R , D ⊆ R n ∖ { 0 } , {\displaystyle \Phi :D\to \mathbb {R} ,\qquad D\subseteq \mathbb {R} ^{n}\smallsetminus \{\mathbf {0} \},}

such that

lim h → 0 Φ ( h ) = 0 and f ( a + h ) − f ( a ) = M ( h ) + Φ ( h ) ⋅ ‖ h ‖ {\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 }

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 ∂ f ∂ x i {\displaystyle {\frac {\partial f}{\partial x_{i}}}} as

f ( a + h ) − f ( a ) = ∑ i = 1 n ∂ f ( a ) ∂ x i h i + Φ ( h ) ⋅ ‖ h ‖ {\displaystyle f(\mathbf {a} +\mathbf {h} )-f(\mathbf {a} )=\displaystyle \sum _{i=1}^{n}{\frac {\partial f(a)}{\partial x_{i}}}h_{i}+\Phi (\mathbf {h} )\cdot \Vert \mathbf {h} \Vert }

Fundamental increment lemma

Differentiability in higher dimensions

See also