determine for what numbers, if any, the given function is discontinuous. f(x) = { x - 2 if x <= 2; x^2 if x…

determine for what numbers, if any, the given function is discontinuous. f(x) = { x - 2 if x <= 2; x^2 if x > 2 } select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the function f is discontinuous at (use a comma to separate answers as needed.) b. the function f is continuous everywhere.

determine for what numbers, if any, the given function is discontinuous. f(x) = { x - 2 if x <= 2; x^2 if x > 2 } select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the function f is discontinuous at (use a comma to separate answers as needed.) b. the function f is continuous everywhere.

Answer

Explanation:

Step1: Find left - hand limit

As (x\to2^{-}), (f(x)=x - 2). So (\lim_{x\to2^{-}}f(x)=\lim_{x\to2^{-}}(x - 2)=2-2 = 0).

Step2: Find right - hand limit

As (x\to2^{+}), (f(x)=x^{2}). So (\lim_{x\to2^{+}}f(x)=\lim_{x\to2^{+}}x^{2}=2^{2}=4).

Step3: Evaluate function at (x = 2)

When (x = 2), (f(2)=2 - 2=0).

Step4: Check continuity

Since (\lim_{x\to2^{-}}f(x)=0), (\lim_{x\to2^{+}}f(x)=4), and (\lim_{x\to2^{-}}f(x)\neq\lim_{x\to2^{+}}f(x)), the function is discontinuous at (x = 2).

Answer:

A. The function f is discontinuous at (2)