I am a bit confused how the formula is different from a simple lerp.
k(x) = 1/x
y_i * k(x - x_i) + y_i+1 * k(x_i+1 - x)
F = --------------------------------------- =
k(x - x_i) + k(x_i+1 - x)
y_i / (x - x_i) + y_i+1 / (x_i+1 - x)
= --------------------------------------- =
1 / (x - x_i) + 1 / (x_i+1 - x)
y_i * (x_i+1 - x) / A + y_i+1 * (x - x_i) / A
= ---------------------------------------------
(x_i+1 - x) / A + (x - x_i) / A
where A = (x - x_i) * (x_i+1 - x)
scaling to x_i+1 - x_i = 1, and using a = (x - x_i)
F = (1 - a) * y_i + a * y_i+1
The only time this wouldn't hold is when A = 0, but then F is not defined anyway. However, the graph of F shown is decidedly not that of the lerp function.
Oh, it seems that a square fell off. It should have been 1/x^2. 1/x works fine for bitmaps, but in 1-dimensional case 1/x indeed turns the whole thing into linear interpolation, you're right.
