In school, I have been tought that sunlight emits all wavelenghts of visible light.
However, shouldn't also the sun have the footprints of its components? Shouldn't it's spectrum be similar to the hydrogen and helium ones? (it isn't, but why?)
And is there something that emits a fully continous spectrum?
Here is the solar spectrum. Despite what you have been taught, it shows a great deal of structure, dominated by lots of dark (but not black) absorption lines. These "Fraunhofer lines" are indeed the signature of what chemical elements (and a few molecules) are present in the solar photosphere - that region in the outer part of the Sun from whence light escapes.
The ability of light of a particular wavelength to escape from a point inside the Sun depends on what atoms are higher up in the atmosphere that might absorb it. If there is a particular atomic transition that is excited by absorbing photons of that wavelength, then such photons can only escape if they are higher up in the Sun's photosphere and emitted from material at cooler temperatures. Since the intensity of the light goes as the fourth power of temperature then the contribution of those wavelelengths to the spectrum is vastly diminished and we see dark absorption lines.
It is relatively straightforward to produce a much smoother spectrum. The Sun's absorption lines are due to the fact that the Sun gets hotter towards its centre. If we look at radiation coming from a body that (i) absorbs all light that is incident upon it and (ii) is at a uniform temperature, then the emission from that object will be a smooth "blackbody" continuum. Such blackbodies can be created by making a small hole into a heated cavity.
However, shouldn't also the sun have the footprints of its components? Shouldn't it's spectrum be similar to the hydrogen and helium ones? (it isn't, but why?)
The light emitted by the sun is heat energy that's being re-radiated from hot gasses. The actual electromagnetic energy generated in the sun's core is in the form of x-rays; it gets absorbed fairly soon, and then takes millions of years to make it out to the part of the sun that we see glowing.
However the sun's spectrum isn't continuous -- it has colder gasses overlaying the glowing parts, and those cold gasses absorb radiation in bands centered around the same spectral lines as they would glow in a gas discharge tube (See the comments for why).
In fact, helium was first discovered as spectral absorption lines in the sun's light, and is named after Helios, the Greek word for "sun".
And is there something that emits a fully continuous spectrum?
Do a web search on "black body radiation". Things that are perfectly black, and are hot, emit radiation in a continuous spectrum (but not one that's flat).
