continuous spectrum
When you play a single sustained note, its sound is built from a few exact frequencies — a discrete spectrum, a handful of vertical lines. When you clap once, the sound is a brief burst that contains a little of every frequency, smeared into a smooth band. That smooth band is a continuous spectrum: the Fourier transform F(k) of a non-periodic signal, viewed as a curve telling you how the signal's content is spread across all frequencies at once, with no gaps and no isolated lines.
The contrast with a Fourier series is the whole point. A periodic function lives on a discrete set of harmonics n times omega_0, so its spectrum is a comb of spikes at those frequencies. As you let the period grow toward infinity, the spacing between adjacent harmonics shrinks to zero, the comb fills in, and the spectrum becomes continuous. In that limit the quantity F(k) is no longer the amplitude of one harmonic; it is a spectral density — amplitude per unit frequency — so what is physical is the area F(k) dk over a band of frequencies, not the value at a single point k. The magnitude |F(k)| (or |F(k)|^2) is the amplitude (or energy) spectrum.
Continuous spectra are how engineers and physicists describe the frequency content of transients, noise, pulses, and light. The spectrum of a single short pulse is broad; the spectrum of a long, slowly varying signal is narrow and concentrated near low frequencies. This reciprocity — short in time means wide in frequency — is the practical face of the scaling property and underlies bandwidth, the resolution limit of optical instruments, and the energy-time uncertainty relation. A genuinely periodic signal does not have an ordinary continuous spectrum; its transform is a sum of Dirac deltas, the mathematical ghost of the original spectral lines.
A decaying pulse f(t) = e^{-a t} for t > 0 (and 0 before) has continuous spectrum F(omega) = 1/(a + i omega), whose magnitude 1/sqrt(a^2 + omega^2) is a smooth hump centered at omega = 0 — broad if a is large, narrow if a is small.
A single transient spreads its energy smoothly over all frequencies; faster decay (larger a) means a wider spectral hump.
Because F(k) is a density, the height at a single frequency carries no energy by itself — reading |F(k)| as the amount at exactly that frequency, rather than per unit frequency, is a common conceptual error.