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.
While technology can and should be used to develop intuition, it does not help much when trying to map intuition to formalism. And that is where the epsilon-delta definition is hard to grasp; continuous is always more subtle/difficult to understand than discrete. This has already been well studied/discussed in mathematics education;
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.