Sampling a Unit step function I have following two related questions concerning unit step function. 

1-I want to sample the following signal. What will be the sample value at t=0? The signal takes 0 time to change from 0 to 1. So I  am confused what should be the sample value at t=0: 0 or 1?      

2- This signal is having infinite bandwidth (This signal is not possible in real world as all physical systems takes certain time to change its values, but here the value is changing in 0 time ). What is its counterpart in discrete domain (again with infinite bandwidth).
1-) The value at t = 0 of an ideal step function converges to $\frac{f(0-) + f(0+)}{2}$ at 0 if you approach the question from a Fourier transform perspective (see Fourier transform of a discontinuous function).

2-) The analogous function in discrete time is still a unit step defined as:
$$
u[n] =
\begin{cases}
0 &amp; \text{if n &lt;  0} \\
1 &amp; \text{if n &gt;= 0} \\
\end{cases}$$

The concept of infinite frequency does not convey to discrete time representation.  In discrete time, frequency is represented by sequences of complex exponentials rather than continuous complex exponential functions.  Complex exponential sequences are not unique for frequencies above 1/2 your discrete time sampling rate so your frequency range is limited to $[0, \frac{f_s}{2}]$, where $f_s$ is your sampling rate.

See sampling theory, the nyquist frequency and a discussion of the discrete Fourier transform for a more detailed discussion.