Question

from the graph of f, state each x - value at which f is discontinuous. for each x - value, determine whether f is continuous from the right, or from the left, or neither. (enter your answers from smallest to largest.)

Explanation:

Step1: Recall continuity definition

A function \(y = f(x)\) is continuous from the right at \(x=a\) if \(\lim_{x
ightarrow a^{+}}f(x)=f(a)\) and continuous from the left at \(x = a\) if \(\lim_{x
ightarrow a^{-}}f(x)=f(a)\). Discontinuities occur where there are jumps, breaks or holes in the graph.

Step2: Analyze the graph

By observing the graph, we look for points where the graph has a break.

Step3: Identify the first - discontinuous point

At \(x=- 2\), the function has a break. As \(x\) approaches \(-2\) from the right, the function values approach the value of the function at \(x = - 2\). So \(f(x)\) is continuous from the right at \(x=-2\).

Step4: Identify the second - discontinuous point

At \(x = 2\), the function has a break. As \(x\) approaches \(2\) from the left, the function values approach the value of the function at \(x = 2\). So \(f(x)\) is continuous from the left at \(x = 2\).

Step5: Identify the third - discontinuous point

At \(x=4\), the function has a break. As \(x\) approaches \(4\) from the left, the function values approach the value of the function at \(x = 4\). So \(f(x)\) is continuous from the left at \(x = 4\).

Answer:

\(x=-2\), continuous from the right
\(x = 2\), continuous from the left
\(x=4\), continuous from the left