Why is the electromagnetic spectrum continuous?
Because electromagnetic waves or photons differ from each other by a continuous parameter, wavelenght, frequency or photon energy.
Let us consider the visible part of spectrum, as an example. Its wavelenght ranges from 350 nanometers to 700 nm. There are infinite different values in the interval, as 588.5924 and 589.9950 nanometers, the two orange-yellow lines emitted by sodium atoms.
As for real numbers, there are also infinite wavelength values in the narrow interval between 588.5924 nm and 589.9950 nm.
In this sense, of a range of possible values of wavelenght, frequency and photon energy, the spectrum is "potentially" continuous.
A really continuous spectrum is emitted by a candle, a glowing wire or furnace. Which means that in a wide interval of energy, all possible electromagnetic radiation are really emitted, with more or less intensity.