# Which conditions must fulfill a time-limited signal (so of unlimited bandwidth) $f(t)$ to have a bounded maximum slew rate?

Which conditions must fulfill a time-limited signal (so of unlimited bandwidth) $$f(t)$$ to have a bounded maximum slew rate?

I want to know about which conditions must fulfill a real-valued time-limited function (so compacted-supported and with unlimited bandwidth), at least in the case of one-variable $$f(t)$$, to have a bounded maximum slew rate $$\max_t \left|\frac{d}{dt}f(t)\right|< \infty$$.

I have already known that band-limited functions have bounded maximum rate of change has is explained here, and also that functions with unlimited bandwidth could achieve and infinite maximum rate of change as I explained on an answer of this question, but also, in the body of the question I also show that there exists time-limited functions as: $$f(t) = \begin{cases} \cos^2 \left( \frac{t \pi}{2} \right) , \, \, |t| \leq 1 \\ 0, \, \, |t|>1 \end{cases}$$ which haves bounded maximum rate of change, so I want to know which conditions must fulfil an unlimited-bandwidth function to have a finite maximum slew rate (let them be named as $$\text{mysterious conditions}\,\mathbb{X}$$), and hopefully found a “good” approximation or upper bound for this maximum rate of change $$\max_t \left|\frac{d}{dt}f(t)\right|$$.

Or conversely, given my limited mathematical knowledge and since I prove myself that time-limited functions with bounded maximum slew rate do exists (it wasn´t obvious for me), at least for me, to assume that "because infinite bandwidth signals could achieve infinite max slew rate" $$\Rightarrow$$ "there is no conditions where under them, time-limited functions could be achieving bounded max slew rates" (so examples are just “happy coincidences”), will be falling into a logical fallacy (I believe is named Hasty generalization). So, if you could prove that these $$\text{mysterious conditions}\,\mathbb{X}$$ are nonexistence, it will be also great, since I could be moving out from trying to solve this problem.

Hope you can answer there to keep anything in one place.

Beforehand, thanks you very much.