Confusion about subtle difference between discrete-time and continuous-time
The problem is that $$e^{j 2 \pi t} \neq 1,$$ except in the case when $t$ is an integer. However, in that case, you have the discrete-time formula: $n$ is assumed to be an integer. In most DSP literature, wherever you see $n$ used as in this case, you can safely assume it's an integer.