Another really basic question... this time regarding integration.

Mathguy15

Given a function f define a new function Sf(x) by summing up all values of f(hj)
where 0 ≤ jh < x. That is, if k is such that kh is the largest below x, then
Sf(x) = h[ f(0) + f(h) + f(2h) + .... + f(kh) ]
We call Sf also the ”integral” or ”antiderivative” of f.

The teacher who wrote the lecture notes I'm reading through gives an example of integration. He evaluates Sf(x) for f(x)=1. I don't understand the first sentence:

We have Sf(x) = 0 for x ≤ h.

Why? Sorry for being such a n00b, but I don't understand. Please help me.

Thanks,
Mathguy

By the way, he verifies that the js in the definition are integers.

pwsnafu

Err, the integral is what you get if you take the limit h -> 0.
Is that word-for-word what is written there?

That doesn't seem right. jh is allowed to equal 0, so the largest integer k such that
0 ≤ kh < x ≤ h is when k=0. So Sf(x) = h f(0) = h.

Edit: Maybe he means x < 0?

Mathguy15

Well, Yes, that is word-for-word, but I think he's doing a "preliminary" definition before the real definition. And I was thinking the same thing, because Sf(x) isn't defined for x<0.