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In my text book, Digital Signal Processing, Principles, Algorithms, and Applications by John Proakis, it asks the question to show how $$\delta(n) = u(n) - u(n-1).$$
I can understand how this is true if $n$ is an integer, but sometimes I see the step function as a continuous line instead of a fixed point at an instant. In that case, is it incorrect to think of $u(n)$ as something continuous?
You are correct that this equality is only true when $n$ is an integer.
Explanation
There are in fact 2 different flavors of the delta function.
The Dirac delta function is defined as $$ \delta (x) = 0 \quad x \neq 0$$ $$ \delta (x) \neq 0 \quad x = 0 $$ $$\int_{-\infty}^{\infty} \delta (x)dx = 1$$ You can sense from at least the third part of this definition that the Dirac delta (impulse function) is a continuous function over all $\mathbb R$. It can also have meaning in $\mathbb C$ and in multiple dimensions. Under this definition, the function is only non-zero exactly at $x=0$. It's 0 everywhere else including $x = 0.5$, $x = 0.00000001$, $x = \pi$, etc. In this case $$u(x)-u(x-1) = rect(x-0.5) \neq \delta (x)$$
The Kronecker delta function is a function of two discrete variables and is defined as $$ \delta_{i,j} = \begin{cases} 1 \quad &i=j \\ 0 \quad &i \neq j \end{cases} $$ A special case of the Kronecker delta function is known as the unit sample function $\delta[n] \equiv \delta_{n,0}$ which comes up in digital signal processing often. This can be defined more simply as $$ \delta[n] = \begin{cases} 1 \quad &n=0 \\ 0 \quad &n \text{ is another integer} \end{cases} $$ You can see from this definition that the Kronecker Delta function and therefore the unit sample function are discrete functions that only have meaning as functions of integers. For example, $\delta[0.5]$ does not make any sense because 0.5 does not belong to an integer set. In this case $$u[n]-u[n-1] = \delta [n]$$ where the unit step function $u[n]$ is a sampled version of the continuous unit step function with sample rate $T$. $$u[n] = u(nT)$$
To alleviate some of the confusion, authors tend to use the variable $n$ when dealing with discrete-time functions, and the variable $t$ when dealing with continuous-time functions. You will also notice from my answer that people also often use brackets instead of parentheses when dealing with discrete functions, although Proakis does not do this. For this question you need to assume $n$ is an integer based on the fact you're being asked to prove the equality.
