Question

Let the function f be defined by \(f(x) = \frac{x-|x|}{x}\), for \(x\neq 0\) and \(f(0) = 2\) then f is

• A. continuous nowhere
• B. continuous for all \(x\) except at \(x = 0\)
• C. continuous everywhere
• D. continuous for all \(x\) except at \(x = 1\)

Hint:

Check the formula for x greater than 0 and less than 0, then compare limits at 0 with f(0).

Answer 1

The Correct Option is B

Step 1: Sketch the graph:
The graph is the line $y=2$ for $x<0$, the point $(0,2)$, and the line $y=0$ for $x>0$.

Step 2: Look for jumps:
There is a jump from 2 to 0 at $x=0$ only. No jump at any other point, including $x=1$.

Step 3: Pick:
Option B.

Final Answer:
There is a jump from 2 to 0 at x = 0 only. \[ \boxed{\text{(B) }\text{continuous for all }x\text{ except at }x=0} \]