As the title suggests, I have a confusion here. In a Systems text I am going through, it mentions of the Unit Impulse as an unbounded signal. Yes, the unit impulse "height" is unbounded, but the "strength" of the signal as the way it is mentioned in many texts, is finite, given by the area of the unit impulse. By that line of thinking it is giving me a bounded signal. What am I missing here? How is the unit impulse interpreted/treated in terms of being an input or an output to a system? Thanks!
Bounded or unbounded in that context always refers to amplitude. E.g., BIBO (bounded-input-bounded-output) stability means that a bounded input signal $x(t)$ satisfying $|x(t)|\le A$ always results in a bounded output $y(t)$ satisfying $|y(t)|\le B$ with some finite positive constants $A$ and $B$.
So the Dirac delta impulse is not well-suited for testing stability of a system. E.g., an ideal integrator's response to an impulse at the input is a unit step, which is of course bounded. Yet, the ideal integrator is unstable because its response to a step is unbounded.
In sum, a Dirac delta impulse is unbounded.