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

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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[Mimi]

How would you figure this out algebraically?

[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?

[RobotNewgate]

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