Q.For a function $f$ to be continuous at $x=a$, which of the following conditions must ALL hold?

Continuity at a point is a precise three-part test, not just any one of these conditions on its own.

All three conditions — $f(a)$ defined, the limit existing, and the limit equalling $f(a)$ — must hold together; missing any one makes $f$ discont …

Why (c) is correct: the definition of continuity at a point $x=a$ requires all three conditions together: (1) $f(a)$ is defined, (2) $\lim_{x \to a} f(x)$ exists (LHL $=$ RHL), and (3) $\lim_{x \to a} f(x) = f(a)$. Only when all three hold simultaneously is $f$ continuous at $a$.

Why the other options are wrong:

• (a) $f(a)$ being defined on its own says nothing about the function's tendency near $a$ — $f(a)$ could be defined as some value while the function approaches a completely different value from either side (a removable discontinuity, discussed elsewhere in this chapter).
• (b) The limit existing on its own says nothing about $f(a)$ — the limit could exist while $f(a)$ is undefined, or exists but does not match the limit, both of which are still discontinuities. …

Students often remember only 'the limit must exist' and forget that the limit must also equal $f(a)$ itself — a function can have a perfectly well-defined limit at a point while stil …