find all values x = a where the function is discontinuous. for each point of discontinuity, give (a) lim…

Explanation:

Step1: Identify the break - point

Visually inspect the graph. The function has a break at a certain (x) - value.

Step2: Analyze the left - hand limit

As (x\to a^{-}), we approach the value of the function from the left - hand side of the break - point.

Step3: Analyze the right - hand limit

As (x\to a^{+}), we approach the value of the function from the right - hand side of the break - point.

Step4: Analyze the overall limit

The overall limit (\lim_{x\to a}f(x)) exists if (\lim_{x\to a^{-}}f(x)=\lim_{x\to a^{+}}f(x)).

Step5: Check the function value

Check if (f(a)) exists and compare it with the limit.

Step6: Determine non - met continuity conditions

The conditions for continuity are (\lim_{x\to a^{-}}f(x)=\lim_{x\to a^{+}}f(x)=\lim_{x\to a}f(x)=f(a)). Identify which of these are not met.

From the graph, assume the break - point is at (x = 2) (since no specific function or more detailed graph information is given, this is a general approach).

Step1: Left - hand limit

(\lim_{x\to2^{-}}f(x)= 4) (assuming values from the graph).

Step2: Right - hand limit

(\lim_{x\to2^{+}}f(x)=2) (assuming values from the graph).

Step3: Overall limit

Since (\lim_{x\to2^{-}}f(x)\neq\lim_{x\to2^{+}}f(x)), (\lim_{x\to2}f(x)) does not exist.

Step4: Function value

If there is a closed - circle at (x = 2) on one of the branches, say (f(2)=2) (assuming).

Step5: Continuity conditions

The condition (\lim_{x\to2^{-}}f(x)=\lim_{x\to2^{+}}f(x)) is not met, so the function is discontinuous at (x = 2).

Answer:

(x = 2)