In derivation of fourier transform, we start with the fourier series coefficients. If we let $T \to \infty$, it's common to say the spacing between consecutive fourier coefficient will approach $0$, and we get a continuous spectrum rather than distinct values.
So basically by letting $\omega_0 \to 0$, $n\omega_0$ becomes $\omega$. My question is - why? Isn't it an intederminate form? Maths is not about pure manipulation of symbols, we cannot magically interpret infinitely closely spaced coefficient as a continuous variable $\omega$. No matter how infinitely close to each other these coefficient will be, there will be places "without" them. We'd like to create a function with the domain of all real numbers, right? This is what we mean by "continuous spectrum". But putting these coefficients infinitely close to each other doesn't mean it's a continuous spectrum...
Source and derivation of fourier transform from fourier series
Yes, you can make the transformation to the continuous transform case rigorous under restricted conditions, but it's rigorous development is more trouble than it's worth. However, the intuition in this argument is worth mentioning when trying to motivate the introduction of the Fourier transform.
The reason the argument has hung around so long is that Fourier came up his transform by using this argument. It's about the only natural and compelling derivation leading the Fourier transform; it motivated Fourier, and it's Fourier's argument. There are reasons to consider the discrete series, but Fourier's reasoning remains the only natural motivation for considering "continuous" integral versions of Fourier expansions, at least at an any reasonable elementary level.