By definition, for a function $f$ to be continuous at a point $x=a$, three specific criteria must be met:

1. Defined at point: $f(a)$ must exist (no holes or asymptotes).
2. Limit exists: ${\lim }_{x\to a}f(x)$ must exist (Left-hand limit = Right-hand limit).
3. Consistency: The limit as $x$ approaches $a$ must exactly equal the function's value at $a$, i.e., ${\lim }_{x\to a}f(x)=f(a)$. This ensures there is no 'jump' at $a$.