I'm still a bit confused from lecture, why do we sample to get discrete function f(x, y), construct a continuous function, and then sample again to get a discrete function? Is it because we need to sample at the texture coordinates (u, v) so the original function doesn't work?

I think it is because we want to sample (u, v) but (u, v) might not be discrete values, so we need to make the continuous function. What I'm a little confused about is what a kernel filter k(x, y) might look like