Item — HN Reader

> the removal of sequences and all the associated theorems from the introductory calculus. Instead, start with limits of functions and the notion of continuity.

Strongly disagree.

Sequences(discrete) and Convergence are vital to understanding Calculus. Only then the idea of converging to a limit from left or right makes intuitive sense. Pair it with a graphical view of secants converging to a tangent(continuous) and you get the idea of instantaneous change however infinitesimal it might be.

You need both discrete and continuous ideas to build intuition before you introduce limits of functions and continuity.

Some books that i have found useful - https://news.ycombinator.com/item?id=49308281

1 comments

I don't disagree. Sequences are important, and the bridge between sequences and functions (Bolzano–Weierstrass theorem, mean value theorem, etc.) is crucial.

But they are not immediately needed to understand the limits.

Try to see how far you can get just with the epsilon-delta formulation of limits of functions.

My point is that the ideas of Sequences/Convergence/Infinity (Heine's definition) provides a better intuition than the epsilon/delta limits of function definition.

The former is discrete so you could literally take any interval and demonstrate how an infinite sequence of real numbers within that interval can converge to a "limit". The student can now understand the idea of a "difference" i.e. a finite change that gets smaller and smaller in concrete terms.

You do the above for x (an independent variable over the above sequence yielding a sequence of delta_x's) and y (a dependent function yielding a sequence of delta_y's).

Now the limit of the sequence of the ratios of the above two sets of differences (i.e. sequence of delta_y/delta_x) can be calculated and defined as the "derivative" i.e. rate of change of one w.r.t. another.

Everything is direct and there is no confusion. They can then easily map the idea of the discrete "difference" to a "differential" in a continuous domain/range and see that the exact same techniques/ideas hold.

I'm in the opposite camp. Most of the calculus foundation actually depends on continuity.

And one thing that has changed recently: mobile phones and zooming. Every child now has an almost instinctive understanding of being able to find the correct zoom level on a map so that some feature can fit completely on the screen.

And now the key insight: you can zoom-in on a continuous function indefinitely.

So the epsilon-delta formulation for the limit of functions becomes almost trivially easy to explain. And you can build from there.

I understand that the notion of continuity in itself requires limits to define it properly. But you can do that _later_. I'm speaking from experience of helping a friend's child understand calculus.