>>11437065If you restrict a function's domain to integers, it effectively becomes a bar graph. The delta function can then be considered the difference between sequential bar heights.
The difference between any two sequential integers is 1, right? So, whatever the difference in the y axis is between sequential bars, it's equal to the difference in the y axis divided by the difference in the x axis, because anything divided by 1 is itself.
From this observation, we can generalize the delta function to other step sizes than just 1. Let's say we increment the x axis by 0.5 each time, and compare sequential bar heights in the resulting bar graph. We'd divide each difference between 0.5, since that's our new step size. So if one bar has height 12, and the next has height 18, the raw difference would be 6, and since our step size is halved, the scaled difference would be 6/(1/2) = 12.
The derivative, then, is the continuous function you get when you generalize the delta function to an infinitesimal step size. It's the limit of scaled delta as step size approaches 0.
Hence the notation: dy/dx.
For any other step size, the generalized delta function would be discrete, just like the function you're calculating the deltas of. But a "discrete" function with a fixed infinitesimal step size is by definition a continuous function.
