consider the function pictured below. describe continuity at each requested point. the function is…

consider the function pictured below. describe continuity at each requested point. the function is continuous at x = 1 because the limit and function value ex. the function is not continuous at x = 4 because the limit and function value ex. the function is not continuous at x = 8 because the function value does not ex.

consider the function pictured below. describe continuity at each requested point. the function is continuous at x = 1 because the limit and function value ex. the function is not continuous at x = 4 because the limit and function value ex. the function is not continuous at x = 8 because the function value does not ex.

Answer

Explanation:

Step1: Recall continuity condition

A function $y = f(x)$ is continuous at $x = a$ if $\lim_{x\rightarrow a}f(x)=f(a)$.

Step2: Analyze $x = 1$

At $x = 1$, the left - hand limit $\lim_{x\rightarrow1^{-}}f(x)$ and the right - hand limit $\lim_{x\rightarrow1^{+}}f(x)$ exist and are equal, and also equal to the function value $f(1)$. So the function is continuous at $x = 1$.

Step3: Analyze $x = 4$

At $x = 4$, there is a hole in the graph. The limit $\lim_{x\rightarrow4}f(x)$ exists (left - hand limit equals right - hand limit), but the function value $f(4)$ is a different point (a filled - in dot at a different $y$ - value). So $\lim_{x\rightarrow4}f(x)\neq f(4)$ and the function is not continuous at $x = 4$.

Step4: Analyze $x = 8$

At $x = 8$, the function has a jump. The left - hand limit $\lim_{x\rightarrow8^{-}}f(x)$ and the right - hand limit $\lim_{x\rightarrow8^{+}}f(x)$ are not equal, and also the function value at $x = 8$ is not well - defined in a way that would make it continuous. So the function is not continuous at $x = 8$.

Answer:

The function is continuous at $x = 1$ because the limit as $x$ approaches $1$ exists and is equal to the function value at $x = 1$. The function is not continuous at $x = 4$ because the limit as $x$ approaches $4$ exists but is not equal to the function value at $x = 4$. The function is not continuous at $x = 8$ because the left - hand and right - hand limits as $x$ approaches $8$ are not equal and the function value is not consistent for continuity.