Tas mus li muaj nuj nqi

Ib tug tas mus li muaj nuj nqi yog ib tug muaj nuj nqi uas tsis muaj "jumps", i.e. ib uas cov nram qab no mob yog txaus siab: me me kev hloov cav ua raws li los ntawm me me kev hloov nyob rau hauv lub duas paub qhov tseem ceeb ntawm cov nuj nqi. Lub teeb ntawm xws li ib tug muaj nuj nqi yog ib tug tas mus los yog tus nkhaus.

Saab laaj ntawm lub taw tes tsis pub tshaj rau ib pawg, yuav tsum tau txiav txim los ntawm kev txwv cov tswv yim, uas yog, tus muaj nuj nqi yuav tsum muaj ib tug txwv nyob rau hauv qhov point, uas yog sib npaug zos rau nws tus nqi ntawm cov kev txwv point.

Thaum cov tej yam kev mob ntawm ib co taw tes, hais cov nuj nqi ntawm lub taw tes ib tug discontinuity, piv txwv li nws ntxiv yog lawm. Nyob rau hauv cov lus ntawm cov kev txwv ntawm tsim kua muag point yuav piav raws li ib tug mismatch nyob rau hauv qhov tseem ceeb ntawm lub tawg point nrog ib tug tsis pub tshaj ntawm ib tug muaj nuj nqi (yog hais tias nws tshwm sim).

discontinuity point tej zaum yuav tshem tau, nws yog tsim nyog los txwv cov hav zoov ntawm zog, tab sis Hloov nrog nws cov nuj nqis ntawm ib tug muab point. Nyob rau hauv cov ntaub ntawv no, ntawm no nws yog tau mus "kho", uas yog mus cuag lub ntsiab txhais ntawm saab laaj.
Ib tug txawv kiag li daim duab emerges yog cov kev txwv ntawm ib tug muaj nuj nqi rau ntawm ib tug muab point tsis muaj nyob. Nws muaj ob lub ntsiab lus ntawm discontinuity:

• thawj zoo - thiab muaj finite txwv ob ntawm lub tug-ib tog, thiab tus nqi ntawm ib tug los yog ob leeg tsis coincide nrog rau cov nqi ntawm cov nuj nqi ntawm ib tug muab point;
• tus thib ob hom, thaum tsis muaj ib tug-ib tog los yog ob qho tib si ntawm cov kev txwv los yog qhov tseem ceeb kev txom.

Thaj chaw ntawm tas mus li zog

• Muaj nuj nqi tau li cov kev tshwm sim ntawm kev laij ua hauj lwm, thiab kuj superposition ntawm tas mus li zog ntawm lawv cov sau yog tseem nruam.
• Muab ib tug tas mus muaj nuj nqi uas yog zoo ntawm ib co taw tes, koj yeej ib txwm nrhiav ib tug txaus me me zej zog nyob rau hauv uas nws yuav khaws nws kos npe rau.
• Ib yam li ntawd, yog hais tias nws muaj nuj nqis nyob rau hauv ob lub ntsiab lus A thiab B yog, ntsig txog, ib tug thiab b, nyob rau ib yog txawv los ntawm b, ces rau lub intermediate cov ntsiab lus nws yuav noj tag nrho cov qhov tseem ceeb ntawm lub luv (ib tug; b). Ntawm no koj yuav ua tau ib qho kev nthuav xaus: yog hais tias koj muab ib tug stretched roj hmab band ntsws thiaj li hais tias nws tsis sag (tseem ncaj), ib qho ntawm nws cov ntsiab lus nyob twj ywm nyob ruaj ruaj. Ib tug geometrically nws txhais tau hais tias yog ib tug ncaj kab dua los ntawm tej yam intermediate point ntawm A thiab B, uas tshuam lub teeb ntawm cov nuj nqi.

Nco ntsoov ib co ntawm tas mus li (nyob rau hauv lub cheeb tsam ntawm lawv txhais) ntawm elementary zog:

• qhov;
• rational;
• trigonometry.

Nruab nrab ntawm ob yuav tsum muaj lub tswv yim nyob rau hauv kev kawm txog zauv - yog tas mus li thiab differentiable - yog inextricably txuas. Suffice nws rov qab hais tias rau differentiable zog koj yuav tsum tau mus yuav ib tug tas mus muaj nuj nqi.

Yog hais tias cov nuj nqi yog differentiable ntawm ib co taw tes, muaj tas mus li. Txawm li cas los, nws yog tsis tsim nyog, yog li ntawd nws derivative yog tas mus li.

Ib qho kev ua uas muaj nyob rau hauv ib tug txheej ntawm tas mus li derivative, belongs rau ib daim chav kawm ntawv ntawm tus zog. Nyob rau hauv lwm yam lus, nws yog - ib tug tsis tu ncua differentiable muaj nuj nqi. Yog hais tias lub derivative muaj ib tug tsawg tus naj npawb ntawm cov ntsiab lus ntawm discontinuity (tsuas yog thawj zoo), lub zoo xws li muaj nuj nqi no yog hu ua piecewise du.

Ib qho tseem ceeb tswvyim ntawm zauv tsom xam yog uniformly nruam muaj nuj nqi, uas yog, nws muaj peev xwm yuav nyob rau hauv tej taw tes ntawm nws sau tib yam tas mus li. Yog li, ib tug uas yog pom nyob rau lub teeb ntawm cov ntsiab lus, es tshaj tej tus neeg.

Yog hais tias peb kho ib tug taw tes, koj tau txais tsis muaj dab tsi lwm tus, raws li lub ntsiab txhais ntawm saab laaj, uas yog, los ntawm lub hav zoov ntawm niaj hnub laaj implies hais tias qhov no yog ib tug tas mus muaj nuj nqi. Feem ntau hais lus, lub converse yog tsis muaj tseeb. Txawm li cas los, raws li Cantor lub theorem, yog hais tias tus muaj nuj nqi yog mus rau lub compact, uas yog, nyob rau ib tug raug kaw luv, ces nws yog uniformly nruam rau nws.