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B. P. Lathi in his book "Principles of Linear Systems and Signals" mentions in the Fourier Transform:

When $x(t)$ is periodic, the spectrum is discrete, and $x(t)$ can be expressed as a sum of discrete exponentials with finite amplitudes. However for an aperiodic signal, the spectrum becomes continuous; that is, the spectrum exists for every value of $\omega$, but the amplitude of each component in the spectrum is zero. The meaningful measure here is not the amplitude of a component of some frequency but the spectral density per unit bandwidth.

I get that when the period of the periodic signal tends to infinity, the Fourier coefficient of the Fourier series tends to zero, at least mathematically makes sense. But, physically, seeing all the continuous spectra of a Fourier transform, this amplitude of each component in the spectrum is zero makes no sense. I can literally see an amplitude. At a particular frequency. How is it zero????? What am I missing?

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You can see the non-zero magnitude of the spectrum, but the interpretation that the signal contains a sinusoidal component at a specific frequency where the magnitude is non-zero is wrong. If there is a sinusoidal component present, then there is a Dirac impulse at the respective frequency. A non-zero Fourier transform at a certain frequency is not sufficient.

The squared magnitude of the Fourier transform is the energy spectral density, and the energy inside a certain frequency band can be obtained by integrating the energy density over the respective interval.

An analogy from statistics is the difference between probability and probability density function (PDF). The values of the probability density function are not probabilities, but probabilities can be obtained from the PDF by integrating it over certain intervals.

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The amplitude you “see” in a spectrum is the result of a narrow band filter, a bin of a DFT, or a pixel at some finite DPI. All of those cover some non-zero bandwidth. So what you see does not have infinitesimal bandwidth (or less).

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