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 at \(x=a\) if \(\lim_{x
ightarrow a^{-}}f(x)=\lim_{x
ightarrow a^{+}}f(x)=f(a)\). Discontinuities occur where this fails.

Step2: Examine the graph for breaks

Look for jumps, holes, or vertical - asymptotes in the graph of \(y = f(x)\).

Step3: Check left - hand and right - hand limits

For each \(x\) - value of discontinuity, determine if the limit as \(x\) approaches from the left (\(\lim_{x
ightarrow a^{-}}f(x)\)) or from the right (\(\lim_{x
ightarrow a^{+}}f(x)\)) exists and is equal to the function value at that point (if defined).

Without seeing the actual graph clearly, assume the following general approach for a function with discontinuities at \(x = x_1,x_2,x_3,x_4\) (in ascending order):
If at \(x = x_1\), the function has a break and \(\lim_{x
ightarrow x_1^{+}}f(x)\) exists and is equal to \(f(x_1)\) (if \(f(x_1)\) is defined in the right - hand sense), then it is continuous from the right.
If \(\lim_{x
ightarrow x_2^{-}}f(x)\) exists and is equal to \(f(x_2)\) (if \(f(x_2)\) is defined in the left - hand sense), then it is continuous from the left.

Answer:

Without the actual graph details, we cannot provide specific \(x\) - values and continuity types. But the process to find the \(x\) - values of discontinuity and their left/right - hand continuity is as described above. You need to identify the \(x\) - values where the graph has breaks, jumps, or holes and then check the left - hand and right - hand limits at those points.