E pili ana i ka notation o nā hana lala

Hoʻohana ʻia nā pale curly i ka notation o ka wehewehe ʻana i ka hana me ka ʻokoʻa o ka hihia. Alualu mākou i ka nīnau maʻalahi inā hiki ke hoʻopau ʻia kēia hōʻike a hiki ke hoʻoliʻiliʻi i ka hana i kahi notation me ka ʻole o ia. ʻO kahi laʻana, ka hana

$$f: \mathbb{R} \to \mathbb{R}, f(x) = \left\{\begin{matrix} 42, & \text{falls } x = 0 \\ x, & \text{sonst} \end{matrix}\right.$$

me ke kōkua o nā hana helu helu ʻehā e hoʻohana ana i ka manawa laina hoʻokahi?


ʻAʻole hiki kēlā a hōʻoia mākou me ke kōkua o ka hoʻomau.

Nānā mākou i ke kaʻina \((x_n)\) me \(x_n = \frac{1}{n}\) . No kēia kaʻina \( \lim_{ n \to \infty } x_n = \lim_{ n \to \infty } \frac{1}{n} = 0\) . Hoʻohui ʻia, \(\lim_{ n \to \infty } f(x_n) = \lim_{ n \to \infty } \frac{1}{n} = 0 \neq 42 = f(0)\) . No laila \(f\) hemo ia ma ke kiko \(x=0\) ia ka hoʻopau loa ʻana.

No ka mea ʻo ka huina a me ka huahana o nā hana hoʻomau e hoʻomau hou ma muli o nā paukū kaulahao, hiki i kekahi ke hana i nā hana hoʻomau me ke kōkua o nā hana helu ʻehā (ʻaʻole loa \(f\) ).

Eia nō naʻe, inā ʻae mākou i ka hana signum discontinuous, no ka laʻana, hiki iā mākou ke loaʻa maʻalahi kahi ʻano inoa. A laila ʻo ia hoʻi

$$f: \mathbb{R} \to \mathbb{R}, f(x) = sgn^2(x-42)+42.$$

No kahi hana maʻamau \(f\) me ka hoʻokaʻawale ʻana i ka hihia

$$f,g,h,a: \mathbb{R} \to \mathbb{R}, f(x) = \begin{Bmatrix} g(x), & \text{falls } a(x) = 0 \\ h(x), & \text{falls } a(x) \neq 0 \end{Bmatrix} = sgn^2 \left(a(x)\right)\cdot h(x) + \left(1-sgn^2\left(a(x)\right)\right)\cdot g(x).$$

Ma ka ʻaoʻao ʻē aʻe, inā ʻoe e nānā i nā hana i nā ʻōlelo papahana, hiki ke hoʻoholo ʻia nā lālā. ʻO kahi laʻana, ma PHP hiki ke palapala ʻia ka hana signum me:

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\(f\) hiki ke hōʻike pū ʻia me ka ʻole inā / ʻē aʻe ka mālama ʻana i nā hale me:

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Inā makemake ʻoe e hana me ka ʻole o ka hoʻohālikelike ʻana o nā mea lawelawe, hiki iā ʻoe ke hele i kahi ʻanuʻu hou aʻe a e hoʻohaʻahaʻa iā ʻoe iho i ka honua nani o nā mea hoʻohana iki:

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Hope