1. The problem asks about the nature of a critical point for a continuous function $f$ on $\mathbb{R}$ at $x = a$. 2. A critical point of a function $f$ is defined as a point where the derivative $f'(x)$ is either zero or undefined. 3. This means at $x = a$, either $f'(a) = 0$ or $f'(a)$ does not exist. 4. Additionally, the derivative from the right $f'(a^+)$ and from the left $f'(a^-)$ may not be equal, which can cause $f'(a)$ to be undefined. 5. Therefore, all the options (a), (b), and (c) describe possible conditions at a critical point. 6. Hence, the correct answer is (d) All the previous.