Let f:I→Rf:I\to\mathbb{R}f:I→R (or C\mathbb{C}C) on an interval I⊆RI\subseteq\mathbb{R}I⊆R. If fff is differentiable on III, one can form f′f'f′. If f′f'f′ is differentiable, one defines the second derivative f′′:=(f′)′.f'' := (f')'.f′′:=(f′)′. Inductively, if f(n−1)f^{(n-1)}f(n−1) is differentiable, the nnnth derivative is ...

Let (X,dX)(X,d_X)(X,dX​) and (Y,dY)(Y,d_Y)(Y,dY​) be metric spaces and let f:E→Yf:E\to Yf:E→Y with E⊆XE\subseteq XE⊆X. The function fff is Hölder continuous on EEE with exponent α∈(0,1]\alpha\in(0,1]α∈(0,1] if there exists a constant C≥0C\ge 0C≥0 such that ∀x,y∈E,dY ⁣(f(x),f(y))≤C dX(x,y)α.\forall x,y\in E,\quad d_Y\!\bigl(f(x),f(y)\bigr)\le C\, d_X(x,y)^{\alpha}.∀x,y∈E,dY​(f(x),f(y))≤CdX​(x,y)α. Any such CCC is called a Hölder constant for (f,α)(f,\alpha)(f,α) on EEE. ...

Let (X,dX)(X,d_X)(X,dX​) and (Y,dY)(Y,d_Y)(Y,dY​) be metric spaces . A function f:X→Yf:X\to Yf:X→Y is a homeomorphism if: fff is bijective , fff is continuous on XXX, and f−1:Y→Xf^{-1}:Y\to Xf−1:Y→X is continuous on YYY. Homeomorphisms are the isomorphisms in topology: they identify spaces that are “the same up to continuous deformation.” Properties preserved by homeomorphisms are called topological invariants (e.g., compactness , connectedness ). ...

Let f:X→Yf:X\to Yf:X→Y be a function. The image (or range) of fff is f(X):={f(x):x∈X}⊆Y.f(X):=\{f(x):x\in X\}\subseteq Y.f(X):={f(x):x∈X}⊆Y. More generally, for a subset A⊆XA\subseteq XA⊆X, the image of AAA under fff is f(A):={f(a):a∈A}⊆Y.f(A):=\{f(a):a\in A\}\subseteq Y.f(A):={f(a):a∈A}⊆Y. The image captures the “actual outputs” of fff and is the natural codomain for which fff becomes surjective (if one replaces YYY by f(X)f(X)f(X)). ...

Let (X,d)(X,d)(X,d) be a compact , connected metric space and let f:X→Rf:X\to\mathbb{R}f:X→R be continuous . Corollary: The set f(X)⊆Rf(X)\subseteq\mathbb{R}f(X)⊆R is a compact interval . In particular, there exist m,M∈Rm,M\in\mathbb{R}m,M∈R with m≤Mm\le Mm≤M such that f(X)=[m,M], f(X)=[m,M], f(X)=[m,M], where m=HAHAHUGOSHORTCODE654s5HBHBXfm=min _X fm=HAHAHUGOSHORTCODE654s5HBHBX​f and M=HAHAHUGOSHORTCODE654s6HBHBXfM=max _X fM=HAHAHUGOSHORTCODE654s6HBHBX​f. ...

Implicit Function Theorem: Let U⊆Rn+mU\subseteq\mathbb{R}^{n+m}U⊆Rn+m be open and let F:U→RmF:U\to\mathbb{R}^mF:U→Rm be of class C1C^1C1. Write points as (x,y)(x,y)(x,y) with x∈Rnx\in\mathbb{R}^nx∈Rn and y∈Rmy\in\mathbb{R}^my∈Rm. Suppose (a,b)∈U(a,b)\in U(a,b)∈U satisfies F(a,b)=0 F(a,b)=0 F(a,b)=0 and the m×mm\times mm×m Jacobian matrix with respect to yyy is invertible at (a,b)(a,b)(a,b): det⁡(∂F∂y(a,b))≠0. \det\left(\frac{\partial F}{\partial y}(a,b)\right)\neq 0. det(∂y∂F​(a,b))=0. Then there exist neighborhoods AAA of aaa and BBB of bbb and a unique C1C^1C1 function g:A→Bg:A\to Bg:A→B such that F(x,g(x))=0for all x∈A. F(x,g(x))=0 \quad \text{for all } x\in A. F(x,g(x))=0for all x∈A. Moreover, g(a)=bg(a)=bg(a)=b and its derivative satisfies Dg(x)=−(∂F∂y(x,g(x)))−1(∂F∂x(x,g(x))). Dg(x)= -\left(\frac{\partial F}{\partial y}(x,g(x))\right)^{-1}\left(\frac{\partial F}{\partial x}(x,g(x))\right). Dg(x)=−(∂y∂F​(x,g(x)))−1(∂x∂F​(x,g(x))). ...

A function g:D→Rmg:D\to \mathbb{R}^mg:D→Rm is implicitly defined by an equation F(x,y)=0F(x,y)=0F(x,y)=0 if there is a map F:U⊆Rn+m→RmF:U\subseteq \mathbb{R}^{n+m}\to \mathbb{R}^mF:U⊆Rn+m→Rm and a set D⊆RnD\subseteq \mathbb{R}^nD⊆Rn such that (x,g(x))∈U(x,g(x))\in U(x,g(x))∈U for all x∈Dx\in Dx∈D and F(x,g(x))=0for all x∈D.F\bigl(x,g(x)\bigr)=0 \quad \text{for all } x\in D.F(x,g(x))=0for all x∈D. ...

An indexed family of sets is a function i⟼Aii \longmapsto A_ii⟼Ai​ from an index set III to the class of sets. It is typically denoted by {Ai}i∈I\{A_i\}_{i\in I}{Ai​}i∈I​. Indexed families let one take unions/intersections over arbitrary index sets (including infinite ones) and are pervasive in analysis (e.g., sequences of sets correspond to the case I=NI=\mathbb{N}I=N). ...

Let (X,≤)(X,\le)(X,≤) be a partially ordered set and let S⊆XS\subseteq XS⊆X. An element i∗∈Xi^\ast\in Xi∗∈X is the infimum of SSS, written i∗=inf⁡Si^\ast=\inf Si∗=infS, if: i∗i^\asti∗ is a lower bound of SSS, i.e. ∀s∈S, i∗≤s\forall s\in S,\ i^\ast\le s∀s∈S, i∗≤s, and i∗i^\asti∗ is the greatest such lower bound: for every lower bound ℓ\ellℓ of SSS, one has ℓ≤i∗\ell\le i^\astℓ≤i∗. Infima are the “best possible” lower bounds. In R\mathbb{R}R, the existence of infima for bounded-below sets is equivalent to the existence of suprema for bounded-above sets. ...

A function f:X→Yf:X\to Yf:X→Y is injective (or one-to-one) if ∀x1,x2∈X, (f(x1)=f(x2)⇒x1=x2).\forall x_1,x_2\in X,\ \bigl(f(x_1)=f(x_2)\Rightarrow x_1=x_2\bigr).∀x1​,x2​∈X, (f(x1​)=f(x2​)⇒x1​=x2​).Injectivity means that outputs uniquely determine inputs (within the domain ). This is the exact condition needed for a (two-sided) inverse function to exist after restricting the codomain to the image . ...