7 Kurtèc kai koðlec sunart seic Parat rhsh An x 1, x, x 2 R, tìte x 1 x x 2 upˆrqei monadikì λ [0, 1] ste x = (1 λ)x 1 + λx 2. Prˆgmati (upojètontac qwrðc blˆbh thc genikìthtac ìti x 1 x 2 ) èqoume x = x 2 x x 2 x 1 x 1 + x x 1 x 2 x 1 x 2 kai 0 x x 1 x 2 x 1 1 an kai mìnon an x 1 x x 2. Gewmetrikˆ (ìtan λ (0, 1)) h sqèsh x = (1 λ)x 1 + λx 2 shmaðnei ìti to x diaireð to eujôgrammo tm ma pou en nei ta x 1 kai x 2 se lìgo x x 1 x 2 x = λ 1 λ. Orismìc 7.1 An I R diˆsthma 1, mia sunˆrthsh f : I R lègetai kurt an gia kˆje x 1, x 2 I kai λ [0, 1], f((1 λ)x 1 +λx 2 ) (1 λ)f(x 1 )+λf(x 2 ) kai koðlh an gia kˆje x 1, x 2 I kai λ [0, 1], f((1 λ)x 1 +λx 2 ) (1 λ)f(x 1 )+λf(x 2 ). H f lègetai gnhsðwc kurt [antðstoiqa, gnhsðwc koðlh] an gia kˆje x 1, x 2 I, x 1 x 2 kai λ (0, 1), f((1 λ)x 1 + λx 2 ) <(1 λ)f(x 1 ) + λf(x 2 ) [antðstoiqa] f((1 λ)x 1 + λx 2 ) >(1 λ)f(x 1 ) + λf(x 2 ). Mia sunˆrthsh f eðnai koðlh an kai mìnon an h f eðnai kurt. Gewmetrikˆ, mia sunˆrthsh f : I R eðnai kurt an kai mìnon an to grˆfhmˆ thc eðnai kˆtw apì kˆje qord tou. Prˆgmati: èstw x 1, x 2 I, x 1 < x 2 kai y i = f(x i ), i = 1, 2. An to shmeðo (x, 0) diaireð to eujôgrammo tm ma 1 UpenjumÐzoume ìti diˆsthma eðnai èna uposônolo I R pou perièqei kˆje eujôgrammo tm ma pou en nei dôo shmeða tou, dhlad pou ikanopoieð x, y I, x < y [x, y] I. 1
pou en nei ta (x 1, 0) kai (x 2, 0) se lìgo x x 1 = λ x 2, tìte apì to Je rhma x 1 λ tou Jal prokôptei ìti to antðstoiqo shmeðo (x, y) thc qord c pou en nei ta shmeða (x 1, y 1 ) kai (x 2, y 2 ) diaireð th qord ston Ðdio lìgo, kai sunep c to (0, y) diaireð to eujôgrammo tm ma pou en nei ta (0, y 1 ) kai (0, y 2 ) pˆli ston Ðdio lìgo. 'Ara y y 1 = λ, y 2 y 1 λ dhlad y = (1 λ)y 1+λy 2 = (1 λ)f(x 1 )+λf(x 2 ). Epomènwc h anisìthta f((1 λ)x 1 + λx 2 ) (1 λ)f(x 1 ) + λf(x 2 ) isodunameð me thn f(x) y. Ja deðxoume ìti mia kurt sunˆrthsh orismènh se anoiktì diˆsthma eðnai pˆnta suneq c, kai, an eðnai paragwgðsimh, h parˆgwgìc thc eðnai aôxousa. L mma 7.1 'Estw f : (a, b) R mia sunˆrthsh. H f eðnai kurt an kai mìnon an, gia kˆje x, y, z (a, b), x < y < z. (1) Apìdeixh An h f eðnai kurt, gia kˆje y (x, z) èqoume y = z y x + y x z z x z x ˆra f(y) f(x) + f(z) (2) isodônama (y>x) () EpÐshc h (2) isodunameð me thn. (z>y) () kai ètsi h (1) apodeðqjhke. AntÐstrofa an h anisìthta (1) isqôei gia kˆje x < y < z sto (a, b), tìte isqôei kai h (2), isodônama (grˆfontac y = λx + (1 λ)z ìpou λ = z y z x ) ˆra h f eðnai kurt. f(λx + (1 λ)z) λf(x) + (1 λ)f(z) ShmeÐwsh H anisìthta (1) onomˆzetai sun jwc {h idiìthta twn tri n qord n}. 2
L mma 7.2 'Estw f : (a, b) R kurt sunˆrthsh. H sunˆrthsh eðnai aôxousa. g x : (a, b)\{x} R me g x (y) = Apìdeixh An a < s < t < u < b apì thn anisìthta (1) èqoume f(t) f(s) t s f(u) f(s) u s f(u) f(t) u t. (3) Jètontac u = x apì to dexiˆ skèloc thc (3) prokôptei 2 ìti a < s < t < x g x (s) g x (t), dhlad h g x eðnai aôxousa sto (a, x). Jètontac s = x apì to aristerˆ skèloc thc (3) prokôptei ìti x < t < u < b g x (t) g x (u), dhlad h g x eðnai aôxousa sto (x, b). Tèloc, jètontac t = x apì thn anisìthta (3) prokôptei ìti a < s < x < u < b g x (s) g x (u) kai sunep c h g x eðnai aôxousa sto (a, x) (x, b). Prìtash 7.3 'Estw I R diˆsthma kai f : I R kurt sunˆrthsh. Tìte h f eðnai suneq c se kˆje eswterikì shmeðo 3 tou I (ìqi ìmwc katanˆgkh sta ˆkra). Apìdeixh 'Estw x o I kai c, d (a, b) ste c < x o < d. Ja deðxoume ìti h f eðnai suneq c sto x o. Apì to L mma 7.2, h sunˆrthsh eðnai aôxousa, ˆra g = g xo : [c, x o ) (x o, d] me g(y) = f(y) f(x o) o g(c) g(y) g(d) gia kˆje y [c, x o ) (x o, d]. 2 parat rhse ìti gx (y) = g y (x), x y 3 dhlad se kˆje xo I gia to opoðo upˆrqoun c, d I me c < x o < d. 3
Sunep c an M max{ g(c), g(d) }, èqoume M g(c) g(y) g(d) M gia kˆje y [c, x o ) (x o, d] opìte f(y) f(x o ) o M f(y) f(x o) M o. 'Epetai ìti an dojeð ε > 0 tìte upˆrqei δ > 0 (mˆlista, δ = ε ) ste gia kˆje M y (a, b) me o < δ na isqôei f(y) f(x o ) < ε, ˆra h f eðnai suneq c sto x o. Parathr seic 7.4 (i) Parat rhse ìti to δ pou epilèxame sthn apìdeixh exartˆtai (ìqi mìnon apì to ε allˆ) kai apì to x o, giatð ta c, d, ˆra kai to M, exart ntai apì to x o. (ii) Mia kurt sunˆrthsh orismènh sèna kleistì kai fragmèno diˆsthma [a, b] eðnai suneq c sto (a, b), ìqi ìmwc katanˆgkh sto [a, b]. 'Ena parˆdeigma eðnai h f : [0, 1] R { x f(x) = 2, x 0 1 x = 0 h opoða eðnai kurt, allˆ den eðnai suneq c sto 0. (iii) Apì to L mma 7.2 den eðnai dôskolo na deðxei kaneðc ìti an f : (a, b) R eðnai kurt sunˆrthsh tìte gia kˆje x (a, b) ta ìria f(t) f(x) lim t x t x = f (x) kai lim s x f(s) f(x) s x = f +(x) upˆrqoun. Den èpetai ìmwc katanˆgkh ìti h parˆgwgoc f (x) upˆrqei. 'Ena parˆdeigma eðnai h f : ( 1, 1) R me f(x) = x. Je rhma 7.5 'Estw f : (a, b) R paragwgðsimh. Tìte Mˆlista f aôxousa f kurt. f gnhsðwc aôxousa f gnhsðwc kurt. 4
Apìdeixh 'Estw ìti h f eðnai kurt. Tìte apì to L mma 7.1 gia kˆje x, y, z (a, b) èqoume kai sunep c x < y < z f (x) = lim y x+ ìtan x < z. EpÐshc ìmwc èqoume kai sunep c ìtan x < z, ˆra telikˆ x < y < z x < z f (x) = lim y x+ lim y z = f (z) lim y z dhlad h f eðnai aôxousa. 'Estw ìti h f den eðnai kurt. Pˆli apì to L mma 7.1, h anisìthta (1) den ja isqôei gia ìlec tic triˆdec x < y < z. Ja upˆrqoun loipìn x, y, z (a, b) me x < y < z ste >. Apì to Je rhma Mèshc Tim c brðskoume ξ (x, y) kai η (y, z) ste f (ξ) = > = f (η) opìte ξ < η allˆ f (ξ) > f (η), ˆra h f den eðnai aôxousa. H deôterh isodunamða af netai wc ˆskhsh. Pìrisma 7.6 'Estw f : (a, b) R duo forèc paragwgðsimh. Tìte f (x) 0 gia kˆje x (a, b) f kurt. Mˆlista an f (x) > 0 gia kˆje x (a, b) tìte h f eðnai gnhsðwc kurt, allˆ to antðstrofo den isqôei pˆnta (p.q. f(x) = x 8 ). = f (z) 5
Apìdeixh H isodunamða èpetai apì to Je rhma kai to gegonìc ìti (efìson h f eðnai paragwgðsimh) h f eðnai mh arnhtik an kai mìnon an h f eðnai aôxousa. Tèloc, an f (x) > 0 gia kˆje x (a, b) tìte h f eðnai gnhsðwc aôxousa sto (a, b), ˆra h f eðnai gnhsðwc kurt. To epìmeno pìrisma prokôptei amèswc an jewr soume thn f. Pìrisma 7.7 'Estw f : (a, b) R paragwgðsimh. Tìte f fjðnousa f koðlh. 'Estw f : (a, b) R duo forèc paragwgðsimh. Tìte f (x) 0 gia kˆje x (a, b) f koðlh. Orismìc 7.2 'Estw f : (a, b) R mia sunˆrthsh. 'Ena shmeðo ξ (a, b) eðnai shmeðo kamp c gia thn f an upˆrqei δ > 0 me (ξ δ, ξ + δ) (a, b) ste h f na eðnai gnhsðwc kurt sto (ξ δ, ξ] kai gnhsðwc koðlh sto [ξ, ξ + δ) h f na eðnai gnhsðwc koðlh sto (ξ δ, ξ] kai gnhsðwc kurt sto [ξ, ξ +δ). Parat rhsh 7.8 An h f : (a, b) R eðnai suneq c paragwgðsimh, èna ξ (a, b) eðnai shmeðo kamp c gia thn f an kai mìnon an eðnai shmeðo topikoô akrìtatou gia thn f. Prˆgmati, an upˆrqei δ > 0 ste h f na eðnai gnhsðwc kurt sto (ξ δ, ξ] kai gnhsðwc koðlh sto [ξ, ξ + δ), tìte (apì to prohgoômeno je rhma kai to pìrisma) h f eðnai gnhsðwc aôxousa sto (ξ δ, ξ) kai gnhsðwc fjðnousa sto (ξ, ξ + δ). Epeid h f eðnai suneq c, èpetai ìti èqei topikì mègisto sto ξ. OmoÐwc, an h f na eðnai gnhsðwc koðlh sto (ξ δ, ξ] kai gnhsðwc kurt sto [ξ, ξ + δ), tìte h f èqei topikì elˆqisto sto ξ. Prìtash 7.9 'Estw f : (a, b) R duo forèc paragwgðsimh. An to ξ eðnai shmeðo kamp c gia thn f, tìte f (ξ) = 0. To antðstrofo den isqôei pˆnta. Parˆdeigma: f(x) = x 4. An ìmwc f (ξ) = 0 kai epiplèon h f (ξ) upˆrqei kai eðnai mh mhdenik, tìte to ξ eðnai shmeðo kamp c gia thn f. Apìdeixh An to ξ eðnai shmeðo kamp c, tìte h f èqei topikì akrìtato sto ξ kai ˆra f (ξ) = 0. An f (ξ) = 0 kai h f (ξ) upˆrqei kai eðnai mh mhdenik, tìte h f èqei topikì akrìtato sto ξ, ˆra to ξ eðnai shmeðo kamp c. 6
Orismìc 7.3 An f : (M, + ) R eðnai mia sunˆrthsh, h eujeða y = ax+b lègetai (plˆgia) asômptwtoc thc f kaj c x + an to lim x + (ax + b f(x)) upˆrqei kai eðnai to 0. AntÐstoiqa orðzetai h asômptwtoc miac sunˆrthshc f : (, M) R kaj c x. An c R eðnai shmeðo suss reushc tou pedðou orismoô miac sunˆrthshc f, h eujeða x = c lègetai katakìrufh asômptwtoc thc f sto c an toulˆqiston èna apì ta ìria lim x c f(x) lim x c f(x) eðnai to + to. 'Askhsh 7.10 Ikan kai anagkaða sunj kh ste h eujeða y = ax + b na eðnai f(x) asômptwtoc thc f kaj c x + eðnai na upˆrqoun ta ìria lim x + = a x kai lim x + (f(x) ax) = b. 7