Question

question 9
the graph below is the function $f(x)$
determine which one of the following rules for continuity is violated first at $x=2$.
$circ$ $f(a)$ is defined.
$circ$ $lim_{x \to a} f(x)$ exists.
$circ$ $lim_{x \to a} f(x) = f(a)$.

Explanation:

Step1: Check if $f(2)$ is defined

From the graph, at $x=2$, there is an open circle, meaning $f(2)$ is not defined.

Step2: Evaluate continuity rules

The three rules for continuity at $x=a$ are:

1. $f(a)$ is defined
2. $\lim_{x \to a} f(x)$ exists
3. $\lim_{x \to a} f(x) = f(a)$

Since the first rule fails, we do not need to check the others.

Answer:

$\boldsymbol{f(a) \text{ is defined.}}$