graph the function. then determine for what numbers, if any, the function is discontinuous. f(x) = { 1 if x…

Explanation:

Step1: Recall continuity definition

A function $f(x)$ is continuous at $x = a$ if $\lim_{x\rightarrow a}f(x)=f(a)$. Consider a prime - number $p$.

Step2: Analyze left - hand and right - hand limits

Let $p$ be a prime number. The left - hand limit $\lim_{x\rightarrow p^{-}}f(x)$ and right - hand limit $\lim_{x\rightarrow p^{+}}f(x)$ do not exist in the usual sense. Because in any open interval around a prime number $p$, there are both prime and non - prime numbers. For example, if we approach $p$ from the left or right, we can find sequences of prime and non - prime numbers that approach $p$. The value of $f(x)$ jumps between $1$ (when $x$ is prime) and $2$ (when $x$ is non - prime). So, $\lim_{x\rightarrow p}f(x)$ does not exist for prime numbers $p$.

Answer:

A. The function is discontinuous at each prime number.