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I would like to know how I can find the sampling rate for the continuous-time signal: u(t) - u(t-2014), where u(t) is the continuous unit step function can be uniquely recovered from its sample.

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Without any prior knowledge about the signal, you can't reconstruct the signal perfectly from its samples because the signal is not band-limited.

If you have prior knowledge, e.g., you know that it is a box function, then you just need three values: the times of the two edges, in your example $t_1=0$ and $t_2=2014$, and the amplitude of the signal.

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Sampling a continuous signal into discrete samples can be cast as finding an appropriate function to convert $x(t)$ ($t\in \mathbb{R}$) into $x[k]$ ($k\in \mathbb{Z}$). Finding functions $f: \mathbb{R}\mapsto \mathbb{Z}$ is easy, but they are generally not inversible, as stated by Matt (one theoretical reason could be that there is not bijection between $\mathbb{R}$ and $\mathbb{Z}$).

However, if you can restrict the subspace or subset of functions in $\mathbb{R}$, (by prior knowledge) then there are inversible solutions, like the Nyquist-Shannon version. Essentially, band-limited functions can (theoretically) be recovered with discrete samples, with something like "two samples per period".

Other frameworks exist: for instance, FRI or finite-rate of innovation provides you with means to perfectly sample a signal made of piece-wise polynomials. They are non-bandlimited signals, hence cannot be retrieved by traditional Nyquist rate sampling. However, FRI method can recover $K$ delta diracs with a minimum sampling rate, or sampled at the rate of innovation, of $2K$. This method extends to integrals of the above, such as a piece-wise constant signal.

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