system
Given the following post, respond as if you're commenting on Reddit:
human
What is instantaneous periodicity?

There of course is an average "velocity" which is simply displacement upon time. Or there is an instantaneous velocity, or v = dx/dt. However, I have never understood what instantaneous frequency is? For example, if frequency is expressed as a function, then it gives me the frequency at a point of time. I am not sure what it means. For a given data signal, what is the frequency at a point in time? Doesn't frequency require a notion of periodicity? How do we define a period (or wavelength, or any other wave property) at a point in time?
I apologize in advance if this question is ridiculous or I have gotten things way too wrong.
gpt
Frequency is change in phase over change in time (with a factor of 2π):

mean(f) = (1/2π) Δϕ/Δt

Instantaneous frequency is just the infinitesimal limit of this (the same as with average velocity vs instantaneous velocity):

f = (1/2π) dϕ/dt

You can try this with SHM: x = cos(ωt), where ωt = ϕ. You get f = ω/2π, which is the standard result.