Question

for the graph above, state all of the x - values on the interval -3,4 where the function is not continuous. for each x - value, provide a mathematical justification for why the function is not continuous.

Explanation:

Step1: Recall continuity conditions

A function $y = f(x)$ is continuous at $x = a$ if $\lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)=f(a)$.

Step2: Analyze $x=- 2$

At $x = - 2$, $\lim_{x
ightarrow - 2^{-}}f(x)
eq\lim_{x
ightarrow - 2^{+}}f(x)$. The left - hand limit as $x$ approaches $-2$ is not equal to the right - hand limit as $x$ approaches $-2$ (the graph has a jump at $x=-2$).

Step3: Analyze $x = 3$

At $x = 3$, $\lim_{x
ightarrow 3^{-}}f(x)
eq\lim_{x
ightarrow 3^{+}}f(x)$. Also, the function has a break in the graph at $x = 3$. The left - hand limit and the right - hand limit of the function as $x$ approaches $3$ are not equal.

Answer:

The function is not continuous at $x=-2$ and $x = 3$. At $x=-2$, the left - hand limit and the right - hand limit are not equal. At $x = 3$, the left - hand limit and the right - hand limit are not equal.