Hobby mathematician here - sorry if my wording is poor.

So basically I'm looking for a series of continuous functionions that has the limit

if floor(x) is even, f(x)=1

if floor(x) is odd, f(x)=-1

So basically a "blocky" version of the sine function, but still continuous.

I know this exists, but I can't remember the formula and I don't even know how to search for it. I toyed with WolframAlpha a bit but it wouldn't compute some things I entered and the only thing I came up with is that

f(x)=sin(x) sin(2x)/2 sin(3x)/3 sin(4x)/4 ...

looks like it's approaching a continuous version of

for 2*pi*k<x<=2*pi*(k 1) and k being an integer

f(x)=(pi/2)-(x/2)

Which is kinda going in the direction I wanted to go but not quite. That's like the saw blade version of the sine but I'm looking for the castle wall version, if you will.