Cause of continuous spectral lines
The spectral lines represent transitions between pairs of discrete energy levels in the atom; an electron is excited by absorbing the energy of a photon and a transition from one of these levels to another occurs as the photon is destroyed. Emission lines occur when the reverse process happens.

If enough energy is absorbed the atom can be ionised and an electron with un-quantised energy can be liberated and moves to a position of complete separation where attractive forces between electron and nucleus is insignificant. As the liberated electron's energy is un-quantised the absorption giving rise to ionisation is continuous. The absorption is truly continuous, no matter how high a spectral resolution is used discrete absorption lines will not be observed.

Black body radiation is continuous emission from a hot body. The analysis of this emission lead Plank in 1901 to propose, for the first time, the idea that the photons were quantised, thus avoiding the 'ultra-violet catastrophe'. He did this by allowing ultra-violet photons to have more energy than visible ones which in turn had more energy that infra-red ones. Nowadays we know that he energy is directly proportional to frequency via  the Planck constant, $E=hv$. 

In the black body, all the numerous energy levels in the body are highly excited and transitions between them continuously occur as the body is being continuously heated. These transition energies overlap one another in energy so that the spectrum is continuous.  

(In molecules there are also discrete energy levels, in the ground state these are due to the vibrational and rotational motion of the molecule.  A molecule may also have electronically excited states and these can also have discrete vibrational and rotational energy levels. Furthermore there are dissociative excited states that have continuous absorption spectra.)