Math Analysis I - Limits of Functions Part 4 (Discontinuity & Continuity)

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A map is piecewise-continuous when it is continuous everywhere except at a finite number of points, at which the discontinuity is either removable or a jump.

2. Math Analysis Limits of Functions 6 of 36R-2] {o} too)l - n , [ 2 ,f :e . g.. u →u-2] {o} too)E - f - p , [ 2 ,U u1-7×21×29XII. if#i * t oD- o fE , 2 EEO b u t lol1)l - l ,Indeed EA =l-l, fl ) {o)E ln =zero?f e tcontinuousist o be t h a tcontinuous i t be t r u em u s tHE> Fo t h a ts u c hi oothe Ix-ElE, orI eoI f /(x) E)- f t EElet end 0=1takeE > o Ix-Elt x - I - othen, EE, lIX o IIf o - o l - o-fix) I(x) =then I I - oi s a tcontinuous2. Math Analysis Limits of Functions 7 of 36E-Miho}¥IR,f: E f (x)→ =• ,study continuityt h e Ej inofJE-ITT, Eevery pointsince i nis a c c u m u l a t i o n point f o r E .o n H EThen, EE E , continuous e tf i sonlya n dif i flim feelLAI =x u e ¥ - ¥ - t e x ttake fineI t of continuousisNevertheless, I h a s adiscontinuity e t z e r o

1. The function f is not continuous at x, if and only if f has a discontinuity at x. (I point)
2. If f is continuous at x, then f is isolated. (E if J E I E is)
3. The function f is continuous at x. (2. Math Analysis Limits of Functions 9 of 36)
4. Algebraic continuity: f tends to g if and only if f and g are continuous. (I F e tend continuous g if f t g continuous set I I E 1 t by algebraic limits of (s t y l e) f if f - f I N t. fi # f g c x I = E S E ) ( x by (E)(t) continuity f t g = (f t g) E ) = left g) then I o t is continuous (f - g) E a tanalogously, is continuous E Z isolated E point i s a n o f then, e t f I continuous is t g 2. Math)

Analysis Limits of Functions 10 of 36

ii) f o g continuous at x if and only if f is continuous at g(x).

Let C be a point in the domain of f o g. Then C is in the domain of g and g(C) is in the domain of f.

If g is continuous at C and f is continuous at g(C), then f o g is continuous at C.

Math Analysis Limits of Functions 11 of 36

iii) f o g continuous at x if and only if f is continuous at g(x).

Let C be a point in the domain of f o g. Then C is in the domain of g and g(C) is in the domain of f.

If g is continuous at C and f is continuous at g(C), then f o g is continuous at C.

Math Analysis Limits of Functions 12 of 36

Those if g o f continuous at x if and only if g is continuous at f(x).

Let C be a point in the domain of g o f. Then C is in the domain of f and f(C) is in the domain of g.

If f is continuous at C and g is continuous at f(C), then g o f is continuous at C.

Math Analysis Limits of Functions 13 of 36

iv) If g o f and f are continuous at x, then g is continuous at f(x).

gllimjcxll-y-ljlf.in#x)(x) = Ix →Ix → ESE)(Thi)) ( xby continuity= g(goy) (E)=then Ea tisgof continuousEZ isolated Epointi s a n o f Ea tisgof continuousthen,2. Math Analysis Limits of Functions 14 of 36Proposition end iff if oneyEe tcontinuousisthru} fix)flxul.irE , I ,c ' →X ue q u i v a l e n t l y,o r⇐ l i m ( L i m Xu)(xu) ff Xu#Ifor=n n e w→ aProofif then1 isolatedE is point,a neventuallyIX u = E,accumulationI2- pointi f o fi s a nbyt h e n , t h e o re m ,i t sfig jetf i reflint fix) == by continuity(lima)f= u - s o2. Math Analysis Limits of Functions 15 of 36GutinuousfunctioA function f is continuous over an interval if it is continuous at every pointnsoninterveeletof the interval. IR, I:[a. Rb)b,b e r →e , Et h a ts u c h continuousj isTheorem ( e x i s t e d )A zero of a real-valued function f is a point c dom f at which the∈function vanishes. (discordant)fle) (b)l e t j c oFc b)l e , (c)t h a t ifsuchE - oEN-N.tn.- The)2. Math

Analysis Limits of Functions 16 of 36

Ideeostuet-proos@b' •¥ •2i f ' be' b't be t2 - 2 - bne nthe tin end@el 02 - =• •Lemme leu-11d u a l endif only is n oProof e n - l - ilif theire n → oleu-el#oe n - e lI on-lsuppose m i o othen n i le n2.

Math Analysis Limits of Functions 17 of 36

Proof f (b)consider the feel t s oo,c a s e(Otb)ji f1 t h e proof- o i scomplete atb(eth) b,:=f2- t oif e,:= e , (bi)fflail t oi o•= be • (ETI) atf,3- b,:-bif f s o e,:-, (bi)fle, If t oi o•= be • bnt oprocedurer e i t e r a t e t h e e n ,up(dustbin)fs u c h t h a t - othen t h e proof completeis2.

Math Analysis Limits of Functions 18 of 36

If twoconsider then o ti t t h e case,is t h e procedure,fromsentences a r i s e n{en} {bu}MONOTONICALLY MONOTONICALLYINCREASING DECREASING,{but{en}, bnbounded: be r e Ie t Eo nbul KQ u ' → ' →, Ibn-en)bu-linenl i mK - l ein= = =b¥-olim K e lby c o n t i n u i t y, bpInneuence(l)

• (linen)flimf(en)-jIo=Of signbpLuneuencej(K)(limb)but-flintelO±=Of signl-ftell(KIl e k, since fck)feel0 I I 0=by fill-fck l-osandwich2.
• Math Analysis Limits of Functions 19 of 36Corollary f ILet the function be continuous on the interval and suppose it admitsxnon-zero limits (finite or infinite) that are different in sign for tending toI f I fthe end-points of . Then has a zero in , which is unique if isI .strictly monotone onProof l a t end-pointsplI n d i c a t e by p It h eh , asEATENTfind. A H - l a fig. N x t - l p,Elpl ,suppose otby permanence signo fI t t a ) , 11×1t h e t oIpl,E I 'Tx s i x t oItch) I - I p land B> b ec-aa sc h o o s elie, IRb ] B IEe,I and j : →E continuousby fib)fleetconstruction =t oby t h e o r e m Z e r o se x i s t e n c e o fo n monotone)(sinceand i n s e c t i v i t y s t r i c t l y7- ! I t e x t - ot h a tX E such2.
• Math Analysis Limits of Functions 20 of 36Corollary IR Ee, B Ig : [ e . BIf ,

continuous→ o u(e) flbl glblif Cele e n dI Ig b)Fxthen l e , f t ) -Fx)t h e ts u c hEProof hlel-j (e)(el-gconsider t o4lb) ( b ) - g l b l t o- jUK) eeyeborebyc o n t i n u o u sh e e l 4lb)then t oby theorem ze ro se x i s t e n c e o fo nb)1- l e , t h a ts u c hx Eh a - f Ix) get-g-(x) texts o =2. Math Analysis Limits of Functions 21 of 36Corollary ( i n t e r m e d i a t e )depending t h e c a s eo nB I →I-Ie, interveelet t h a tsuchi Ib oot± eo o- IRIf : continuous→(int f )then, (I)f , fstep cNamely, the function f assumed all the values between its infinum and itssupremum r¥5 f# t o oµ i n # IR,2T)( 0 ,f :e. xX i a→ S i ug.. I ]- E l ,2T)l o ,fThe only rule in this kind of statements is ÷t¥¥.that when the domain of the function fis closed and bounded, the range is closedand bounded too. so:*.:*."- r t . .o r e .- #next]-lining,[ i n t f ]R e n j e f , s u p2. Math Analysis Limits of Functions 22 of 36Proof depending o

In the case I consider, I'll show that for any x, there exists a t such that F(c) is such by definition of infimum, F(x) is such by definition of supremum, F(x) is such - h defines C(h) as the function of x, if f is continuous, 1 is T(x, x), since g(x, F(c)).