Continuity over an interval | Limits and continuity | AP Calculus AB | Khan Academy

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A function Ä is continuous over the open interval (a,b) iff it's continuous on every point in (a,b). Ä is continuous over the closed interval [a,b] iff it's continuous on (a,b), the right-sided limit of Ä at x=a is Ä(a) and the left-sided limit of Ä at x=b is Ä(b).
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[@Mimi127]

How would you figure this out algebraically?

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[@user-ll7ui5ym5b]

Can anyone help me if
(negative infinity, 2] is continuous or not?

[@johnferentoulaloum]

I need help in one question. There is an exercise that asks to prove this. Suppose f satisfies the conclusion of the middle value theorem (that in an interval [a,b] f takes every value between f(a) and f(b))and f takes on each value only once (suppose it means a 1-1 function) prove that the function is continuous at [a,b]. Ok it had an ε,δ contradiction proof which made no sense at all. Because it used the middle value theorem WITHOUT knowing that the function is continuous.Circular. Fact is the exercise looks totally wrong, because I can find any function of this kind which is NOT continuous and qualifies. You only have to find a function with no limit at certain x es. (Different - and + limit a x1,x2 etc). And it STILL is 1-1 and takes ALL VALUES between f(a) and f(b).It is a very interesting calculus book that I am studying but either I am missing something or there is a huge mistake. Any help?

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[@loona8398]

Why last one is not continues?

[@Trendy_Frogg]

Becaude the limit of f(x) as x approaches -2 from the right needs to equal the value of f(x) when x is -2, in this case the limit from the right is -3 and the value of the function at x equals -2 is equal to 0 and since they arent equal, It is NOT continues over that closed interval (hope you understand)

[@parameshwarballu1278]

Clear and To the point!

[@lucasargandona4658]

Would (3,4) be continuous?

[@echodiasenemb4611]

Yes... i think

[@uwish5356]

@sxzm isn't an infinite discontinuity considered discontinuous?

[@uwish5356]

@sxzm wait my answer in my exer just change bc everyone is like telling me that it's still continuous bc of the open interval. since it's not really going to 4 just the points between 3 and 4.

[@uwish5356]

@sxzm the answer sheets is still not released. i'll update if it's already released and anyone is still interested with this conversation

[@lestersolaina6734]

@@uwish5356 what is the answer?

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What graph is this

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What function did you use

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