What kind of signal is the signum function? A power signal or an energy signal?

Question

I'm studying "signals and systems". We can indentify two kinds of signals: power signal and energy signal.

The definitions are:

Energy of a time-continous signal $$ E_x = \int_{-\infty}^{+\infty}{|x(t)|}^2dt $$

Power of a time-continous signal $$ P_x = \lim_{Z\rightarrow+\infty}{\frac{1}{2Z}\int_{-Z}^{+Z}{|x(t)|}^2dt}$$

An energy signal satisfies $0<E_x<+\infty$ and $P_x=0$.

A power signal satisfies $0<P_x<+\infty$ and $E_x = +\infty$.

Now, if we have a signum function

$$sgn(t)=\left\{\begin{matrix} 1 & if\;t \ge 0 \\ -1 & if\;t < 0 \end{matrix}\right.$$

and we calculate the formulas, then we obtain:

$$P_x = 0 \;\; E_x=0 $$

My question is, as the title said: the signum function which kind of signal is?

Answer 1

First point is that signum is not continuous with the jump at $t=0$.

Second because of the absolute value in the integral it has the same $E$ and $P$ as a constant signal $1$.

This means that $E_{sgn}=+\infty$ and $P_{sgn}=1$