According to the Shannon Sampling Theorem, you can completely reconstruct a continuous-time signal from discrete, equally spaced samples if the highest frequency in the time signal is less than half the sampling frequency. Half the sampling frequency equals the Nyquist frequency. The Shannon Sampling Theorem bridges the gap between continuous-time signals and digital-time signals.
In practical, signal-sampling applications, digitizing a time signal results in a finite record of the signal, even when you carefully observe the Shannon Sampling Theorem and sampling conditions. Even when the data meets the Nyquist criterion, the finite sampling record might cause energy leakage, called spectral leakage. Therefore, even though you use proper signal acquisition techniques, the measurement might not result in a scaled, single-sided spectrum because of spectral leakage. In spectral leakage, the energy at one frequency appears to leak out into all other frequencies.
Spectral leakage results from an assumption in the fast Fourier transform (FFT) and discrete Fourier transform (DFT) algorithms that the time record exactly repeats throughout all time. Thus, signals in a time record are periodic at intervals that correspond to the length of the time record. When you use the FFT or DFT to measure the frequency content of data, the transforms assume that the finite data set is one period of a periodic signal. Therefore, the finiteness of the sampling record results in a truncated waveform with different spectral characteristics from the original continuous-time signal, and the finiteness can introduce sharp transition changes into the measured data. The sharp transitions are discontinuities. The following figure shows discontinuities.
The discontinuities shown in the previous figure produce leakage of spectral information. Spectral leakage produces a discrete-time spectrum that appears as a smeared version of the original continuous-time spectrum.