What's the essential bandwidth of the unit step function? The Fourier spectrum is in the Figure, how to find the essential bandwidth?
there can be no essential bandwidth of the unit step function.  defining such would lead you to self-contradiction.

suppose you have a function $x(t)$ which has Fourier Transform

$$ X(f) = \int\limits_{-\infty}^{+\infty} x(t) e^{-j 2 \pi f t} \ dt $$

and from $X(f)$, you defined some consistent measure of bandwidth $B_x$ in such a way that is independent of the amplitude of $X$ and is dimensionally consistent with $f$.

now suppose you scale the time argument of $x(t)$ to speed it up or slow it down:

$$ y(t) = x(a t) \quad \text{for } a &gt; 0 $$

we know that the the Fourier Transform of $y(t)$ would be

$$ Y(f) = \frac{1}{a} X\left(\frac{f}{a} \right) $$

and the bandwidth of $Y(f)$ defined in the same manner as $B_x$ would be

$$ B_y = a B_x $$

so, suppose $a&gt;1$, then you speed up $x(t)$ by the factor $a$ and the bandwidth is increased by the same factor.

but if you "speed up" (or slow down) the unit step function, you still have the very same unit step function:

$$ u(t) = u(a t) \quad \text{for } a &gt; 0 $$

where

$$ u(t) \triangleq \begin{cases}
1, &amp; \text{if } t &gt; 0 \\
0, &amp; \text{if } t &lt; 0 \\
\end{cases} $$ 

so scaling time for the unit step does not change it, yet if it had a bandwidth, scaling the time must change that bandwidth.  a contradiction results.