"Continuous" means ongoing in time (as in "continuing"). "Contiguous" means "touching on a boundary". So a "contiguous data structure" is one that "occupies a single chunk of memory and its contents are stored in adjacent memory blocks."
Continuity would only really mean anything for time-series data.
> a. Characterized by continuity; extending in space without interruption of substance; having no interstices or breaks; having its parts in immediate connection; connected, unbroken.
> 1673 N. Grew Idea Phytol. Hist. ii. iii. 65 It is compounded of two Bodies. The one parenchymous; continuous throughout; yet somewhat pliable without a solution of its continuity.
> 1704 I. Newton Opticks ii. ii. 41 The dark intervals must be diminished, until the neighbouring Rings become continuous, and are blended.
> 1796 R. Southey Joan of Arc vii. 6 Round the city stretch'd Their line continuous, massy as the wall Erst by the fearful Roman..raised.
> 1859 C. Darwin Origin of Species xi. 352 In most cases the area inhabited by a species is continuous.
> 1879 J. N. Lockyer Elem. Lessons Astron. vi. 228 If we light a match and observe its spectrum, we find that it is continuous—that is, from red through the whole gamut of colour to the visible limit of the violet.
> 1881 J. C. Maxwell Treat. Electr. & Magnetism (ed. 2) I. 6 Without describing a continuous line in space.