Real Analysis | Topological continuity

We prove that the standard definition for the continuity of a function on the real line is equivalent to the topological definition involving open sets.

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[paradoxicallyexcellent]

The topological definition of continuity is deceptively simple and it takes quite a while to wrap one's head around why it corresponds to (or in fact generalizes) the analytical definition. My favorite way to understand it is to consider the topological definition of a not-continuous function: there exists an open set U such that f^(-1)(U) is not open. For a set to be not open, it has to contain a point which is not an interior point. This not-interior point can be thought of as a point of discontinuity, and lends itself to drawing a picture.

We then think of continuity as lacking any such points of discontinuity.

[myaccount]

Notice that the preimage is not the same as the inverse.
The inverse of a function only exists if it's a bijection, while the preimage always exists.

[goodplacetostop]

9:50

Anyway, have a good rest of the morning, afternoon, evening, wherever you are.

[kartikraturi]

You are very good teacher! You explain so good! Thank you!

[tomatrix]

It’s actually sometimes hard to find the intuition to this theorem but I finally understand it intuitively.

[MonsieurSeize]

By regarding the definition you gave for the limit and for the continuity then the existence of a limit of a function at a point in which it's defined and the continuity of the same fuction at this point are equivalent.

[jonathanjacobson]

I wonder why, apart from possible historical reasons, Topology & Calculus are not taught in conjunction.

[rahgeer.]

I am confused at 4:28
We should have taken the intersection of delta neighborhood and U there, but if we do that then proving delta neighborhood inside 'U' is not that way😢.

[get]

Nice exposition: close points go to close points.

[raghavsomani]

Thanks! Does the definition generalize to any general Topology?

[Reliquancy]

Can an open set be a finite segment of the real line? It seems like you could take a point in the segment then find a point smaller in the neighborhood of that point then find a point smaller than that point in its neighborhood and repeat off to infinity... Or does not including the exact endpoints allow that and still limit the length?