I had always thought of time and frequency as being two different (yet complete) descriptions of the same system, so to me, the phrase "instantaneous frequency" didn't make sense -- frequency is a global description of a function, whereas time is local description of a function.
However, I just got confused by something.
Let's say I have the function $x(t) = \sin(t^2)$. I believe I can say that the frequency of this function increases over time, and, at least intuitively, this makes sense.
But mathematically, I can't make sense of this. Sure, I could take a limit, but I feel like that would be changing the meaning of "frequency" altogether, since it's no longer a description of the actual function.
Is it actually sensible to say that a function's frequency "changes" over time? What is the proper way to state something like this more mathematically?