Let f:E→Rf:E\to\mathbb{R}f:E→R (or C\mathbb{C}C) where E⊆RE\subseteq\mathbb{R}E⊆R is unbounded above. We say lim⁡x→∞f(x)=L\lim_{x\to\infty} f(x)=Lx→∞lim​f(x)=L if ∀ε>0, ∃M∈R such that ∀x∈E, (x>M⇒∣f(x)−L∣<ε).\forall \varepsilon>0,\ \exists M\in\mathbb{R}\ \text{such that}\ \forall x\in E,\ \bigl(x>M \Rightarrow |f(x)-L|<\varepsilon\bigr).∀ε>0, ∃M∈R such that ∀x∈E, (x>M⇒∣f(x)−L∣<ε).Limits at infinity formalize the long-range behavior of functions and are used in asymptotics, improper integrals, and series tests. ...

Let (X,d)(X,d)(X,d) be a metric space and let (xn)(x_n)(xn​) be a sequence in XXX. A point x∈Xx\in Xx∈X is the limit of (xn)(x_n)(xn​) if ∀ε>0, ∃N∈N such that ∀n≥N, d(xn,x)<ε.\forall \varepsilon>0,\ \exists N\in\mathbb{N}\ \text{such that}\ \forall n\ge N,\ d(x_n,x)<\varepsilon.∀ε>0, ∃N∈N such that ∀n≥N, d(xn​,x)<ε. One writes x=lim⁡n→∞xnx=\lim_{n\to\infty}x_nx=limn→∞​xn​ or xn→xx_n\to xxn​→x. ...

Let (X,d)(X,d)(X,d) be a metric space and let A⊆XA\subseteq XA⊆X. A point x∈Xx\in Xx∈X is a limit point (also called an accumulation point or cluster point) of AAA if ∀r>0, (B(x,r)∖{x})∩A≠∅\forall r>0,\ \bigl(B(x,r)\setminus\{x\}\bigr)\cap A \neq \varnothing∀r>0, (B(x,r)∖{x})∩A=∅ (see open ball ). Limit points are the points that can be approached by elements of AAA distinct from the point itself. They determine closedness (a set is closed iff it contains all its limit points) and appear throughout analysis. Compare with isolated points . ...

Let (an)(a_n)(an​) be a sequence in the extended real line [−∞,∞][-\infty,\infty][−∞,∞]. Define the tail suprema sn:=sup⁡{ak:k≥n}∈[−∞,∞].s_n := \sup\{a_k : k\ge n\}\in[-\infty,\infty].sn​:=sup{ak​:k≥n}∈[−∞,∞]. Then the limit superior of (an)(a_n)(an​) is lim sup⁡n→∞an:=lim⁡n→∞sn,\limsup_{n\to\infty} a_n := \lim_{n\to\infty} s_n,n→∞limsup​an​:=n→∞lim​sn​, where the limit exists in [−∞,∞][-\infty,\infty][−∞,∞] because (sn)(s_n)(sn​) is nonincreasing. ...

Let VVV and WWW be vector spaces over the same field F\mathbb{F}F (typically F=R\mathbb{F}=\mathbb{R}F=R or C\mathbb{C}C). A function T:V→WT:V\to WT:V→W is a linear map (or linear transformation) if for all x,y∈Vx,y\in Vx,y∈V and all scalars α,β∈F\alpha,\beta\in\mathbb{F}α,β∈F, T(αx+βy)=αT(x)+βT(y).T(\alpha x+\beta y)=\alpha T(x)+\beta T(y).T(αx+βy)=αT(x)+βT(y).Linear maps are the morphisms in linear algebra; in analysis they model derivatives (as best linear approximations) and bounded linear operators (when norms are present). See also operator norm . ...

Let f:[a,b]→Rf:[a,b]\to\mathbb{R}f:[a,b]→R and let α,β:[a,b]→R\alpha,\beta:[a,b]\to\mathbb{R}α,β:[a,b]→R be functions of bounded variation . Suppose the Riemann–Stieltjes integrals ∫abf dα\int_a^b f\,d\alpha∫ab​fdα and ∫abf dβ\int_a^b f\,d\beta∫ab​fdβ exist. Let c,d∈Rc,d\in\mathbb{R}c,d∈R and define a new integrator γ=cα+dβ. \gamma=c\alpha+d\beta. γ=cα+dβ. ...

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 Lipschitz continuous on EEE if there exists a constant L∈[0,∞)L\in[0,\infty)L∈[0,∞) such that ∀x,y∈E, dY ⁣(f(x),f(y))≤L dX(x,y).\forall x,y\in E,\ d_Y\!\bigl(f(x),f(y)\bigr)\le L\, d_X(x,y).∀x,y∈E, dY​(f(x),f(y))≤LdX​(x,y). Any such LLL is called a Lipschitz constant for fff on EEE. ...

Let U⊆RnU\subseteq\mathbb{R}^nU⊆Rn be open and let f:U→Rnf:U\to\mathbb{R}^nf:U→Rn be $C^1$ . Suppose a∈Ua\in Ua∈U and det⁡Df(a)≠0\det Df(a)\neq 0detDf(a)=0. Corollary: There exist open neighborhoods U0U_0U0​ of aaa and V0V_0V0​ of f(a)f(a)f(a) such that f:U0→V0 f:U_0\to V_0 f:U0​→V0​ is a C1C^1C1 diffeomorphism (bijective , C1C^1C1, with C1C^1C1 inverse). ...

Let F:U⊆Rn+m→RmF:U\subseteq\mathbb{R}^{n+m}\to\mathbb{R}^mF:U⊆Rn+m→Rm be $C^1$ , and suppose (a,b)∈U(a,b)\in U(a,b)∈U satisfies F(a,b)=0F(a,b)=0F(a,b)=0 and det⁡(∂F∂y(a,b))≠0. \det\left(\frac{\partial F}{\partial y}(a,b)\right)\neq 0. det(∂y∂F​(a,b))=0. Corollary: There exist neighborhoods AAA of aaa and BBB of bbb and a C1C^1C1 function g:A→Bg:A\to Bg:A→B such that the solution set of F(x,y)=0F(x,y)=0F(x,y)=0 in A×BA\times BA×B is exactly the graph of ggg: {(x,y)∈A×B: F(x,y)=0}={(x,g(x)): x∈A}. \{(x,y)\in A\times B:\ F(x,y)=0\}=\{(x,g(x)):\ x\in A\}. {(x,y)∈A×B: F(x,y)=0}={(x,g(x)): x∈A}. ...

Let f:E→Rf:E\to\mathbb{R}f:E→R with E⊆XE\subseteq XE⊆X where (X,d)(X,d)(X,d) is a metric space , and let a∈Ea\in Ea∈E. The point aaa is a local maximum of fff if there exists r>0r>0r>0 such that for all x∈E∩B(a,r)x\in E\cap B(a,r)x∈E∩B(a,r), f(x)≤f(a).f(x)\le f(a).f(x)≤f(a). The point aaa is a local minimum of fff if there exists r>0r>0r>0 such that for all x∈E∩B(a,r)x\in E\cap B(a,r)x∈E∩B(a,r), ...