Why not include as a requirement that all functions must be continuous to be differentiable?

Definitions tend to be minimalistic, in the sense that they don't include unnecessary/redundant information that can be derived as a consequence.

Same reason why, for example, an equilateral triangle is defined as having all sides equal, rather than having all sides and all angles equal.

Because that suggests that there might be functions which are discontinuous at $a$ for which it is still true that the limit$$\lim_{t\to0}\frac{f(a+t)-f(a)}t$$exists. Besides, why add a condition that it always holds?

Because then you would need to check continuity for no good reason every time you want to check for differentiability. Besides, it gives the wrong impression of being necessary to include.