Which of the following points is a location where the function $f\left(x\right)=\frac{1}{x-2}$ is discontinuous?

• A. $x=1$
• B. $x=2$
• C. $x=0$
• D. $x=3$

Answer: (B)

Verified step by step guidance

1. Step 1: Recall the definition of discontinuity. A function is discontinuous at a point if it is not defined at that point or if there is a break, jump, or infinite behavior in the graph of the function.
2. Step 2: Analyze the given function f(x) = $\frac{1}{x-2}$. The denominator x - 2 must not be equal to zero because division by zero is undefined.
3. Step 3: Solve the equation x - 2 = 0 to find the value of x where the function is undefined. This gives x = 2.
4. Step 4: Check the other points provided in the options (x = 1, x = 0, x = 3). Substitute these values into the denominator x - 2 to verify that the function is defined at these points.
5. Step 5: Conclude that the function is discontinuous at x = 2 because the denominator becomes zero, making the function undefined at this point.