# What is the relationship between the discrete time and continuous time variables?

While going through Proakis's Digital Signal Processing (page 21) ,he stated that if a continuous time signal $$~x(t)~$$ that has been sampled each $$~T~$$ seconds to produce a discrete time signal $$~x(n)~$$ then the relationship between the variables $$t$$ and $$n$$ is :

$$t=nT \tag($$

Question : in the LHS we have a continuous variable whereas in the RHS we have a variable that can only take step sizes of $$T$$ , so clearly $$t$$ and $$nT$$ do not span the same range , then how is the formula above justified ?

It's about the relationship between the discrete time signal $$x_d[n]$$ and the continuous-time signal $$x_c(t)$$:
$$x_d[n]=x_c(nT)\tag{1}$$
So you formally replace the variable $$t$$ by $$nT$$ but this just means that you sample the continuous-time signal at sample instants $$t_n=nT$$. So $$t=nT$$ is only true for the values of $$t$$ that we're interested in, and these are the discrete values $$t_n=nT$$.