Let n,mβN. Let EβRn, let f:EβRm, let aβE, and write f=(f1β,β¦,fmβ). We say that f is continuous at a if for every Ξ΅>0 there exists Ξ΄>0 such that for every point x=(x1β,β¦,xnβ)βE, ifi=1βnβ(xiββaiβ)2<Ξ΄2,thenj=1βmβ(fjβ(x)βfjβ(a))2<Ξ΅2.
Let nβN and let UβRn. We say that U is open in Rn if for every point x=(x1β,β¦,xnβ)βU there exists a real number r>0 such that every point y=(y1β,β¦,ynβ)βRn satisfyingi=1βnβ(yiββxiβ)2<r2also belongs to U.
Every \reftext{def:bounded-sequence-real-c54-2026a}{bounded sequence} of real numbers has a \reftext{def:subsequence-real-c54-2026a}{subsequence} that converges to a real number in the sense of \ref{def:limit-sequence-real-c54-2026a}.
Let (xnβ)n=1ββ be a sequence of real numbers. A subsequence of (xnβ) is a sequence of the form (xnkββ)k=1ββ, where (nkβ)k=1ββ is a strictly increasing sequence of positive integers.
Let a,bβR with a<b, and let f:[a,b]βR be \reftext{def:continuous-at-point-c54-2026b}{continuous} at every point in [a,b] and \reftext{def:derivative-interior-point-c54-2026b}{differentiable} at every point in (a,b). Assume that f(a)=f(b). Then there exists cβ(a,b) such thatfβ²(c)=0.
Let a,bβR with a<b, and let f:[a,b]βR. Suppose that f has a local \reftext{def:local-extremum-at-point-1d-2026a}{extremum} at an interior point cβ(a,b) and that f is \reftext{def:derivative-interior-point-c54-2026b}{differentiable} at c. Thenfβ²(c)=0.
Let a,bβR with a<b, and let f:[a,b]βR be \reftext{def:continuous-at-point-c54-2026b}{continuous} at every point in [a,b]. Then there exist points xminβ,xmaxββ[a,b] such thatf(xminβ)β€f(x)β€f(xmaxβ)forΒ allΒ xβ[a,b].
Let I be an interval in the sense of \ref{def:interval-real-line-c54-2026a}, let f:IβR, and let cβI. One says that f has a local extremum at c if either f has a local maximum at c or f has a local minimum at c; that is, there exists Ξ΄>0 such that for every xβI with β£xβcβ£<Ξ΄, either f(x)β€f(c) for all such x or f(x)β₯f(c) for all such x.
Every \reftext{def:cauchy-sequence-real-c54-2026a}{Cauchy sequence} of real numbers converges to a real number in the sense of \ref{def:limit-sequence-real-c54-2026a}.
A sequence (anβ)n=1ββ of real numbers is called a Cauchy sequence if for every Ξ΅>0 there exists NβN such that for all integers m,nβ₯N,β£anββamββ£<Ξ΅.
Let (anβ)n=1ββ be a sequence of real numbers, and let LβR. One says that (anβ) converges to L if for every Ξ΅>0 there exists NβN such that for all integers nβ₯N,β£anββLβ£<Ξ΅.In that case one writes limnβββanβ=L or anββL as nββ.
Let a,bβR with a<b. If f:[a,b]βR is \reftext{def:continuity-closed-interval-c54-2026b}{continuous on [a,b]}, then f is \reftext{def:uniform-continuity-real-subset-c54-2026a}{uniformly continuous} on [a,b].
Let f:[a,b]βR, and letP:a=x0β<x1β<β―<xnβ=bbe a \reftext{def:partition-closed-interval-c54-2026a}{partition} of [a,b]. For each i=1,β¦,n, letMiβ=sup{f(t):tβ[xiβ1β,xiβ]},miβ=\reftextdef:lowerβboundβinfimumβc54β2026ainf{f(t):tβ[xiβ1β,xiβ]}.The upper sum and lower sum of f with respect to P are defined byU(f,P)=i=1βnβMiβ(xiββxiβ1β),L(f,P)=i=1βnβmiβ(xiββxiβ1β).
The real numbers have the least upper bound property: whenever SβR is nonempty and \reftext{def:upper-bound-supremum-c54-2026b}{bounded above}, there exists a number uβR such that u=supS.
Let SβR. A number ββR is called a lower bound of S if ββ€s for every sβS.
If S is nonempty and bounded below, then a number mβR is called the greatest lower bound, or infimum, of S if:(i)Β mΒ isΒ aΒ lowerΒ boundΒ ofΒ S,and(ii)Β everyΒ lowerΒ boundΒ βΒ ofΒ SΒ satisfiesΒ ββ€m.In that case one writes m=infS.
Let EβR and let f:EβR. The function f is said to be uniformly continuous on E if for every Ξ΅>0 there exists Ξ΄>0 such that for all x,yβE, if β£xβyβ£<Ξ΄, thenβ£f(x)βf(y)β£<Ξ΅.
Let a,b,cβR satisfy aβ€bβ€c, and let f:[a,c]βR be \reftext{def:riemann-integrable-closed-interval-c54-2026b}{Riemann integrable} on [a,c]. Then the restrictions fβ£[a,b]β:[a,b]βR and fβ£[b,c]β:[b,c]βR are Riemann integrable on [a,b] and [b,c], respectively, andβ«acβf(t)dt=β«abβf(t)dt+β«bcβf(t)dt.In particular,β«acβf(t)dtββ«abβf(t)dt=β«bcβf(t)dt.
Let a,bβR with a<b, let f:[a,b]βR, let P=(x0β,x1β,β¦,xnβ) be a \reftext{def:partition-closed-interval-c54-2026a}{partition} of [a,b], and let (t1β,β¦,tnβ) determine a \reftext{def:tagged-partition-closed-interval-c54-2026a}{tagged partition} of [a,b] relative to P. The corresponding Riemann sum of f is the real number βi=1nβf(tiβ)(xiββxiβ1β).
Let a,bβR with a<b, and let P=(x0β,x1β,β¦,xnβ) be a \reftext{def:partition-closed-interval-c54-2026a}{partition} of [a,b]. A tagged partition of [a,b] relative to P is a choice of points tiββ[xiβ1β,xiβ] for each i=1,β¦,n.