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: Rewrite the function:
For \(x>0\): \(|x|=x\), so \(f(x)=\dfrac{x-x}{x}=0\). For \(x<0\): \(|x|=-x\), so \(f(x)=\dfrac{2x}{x}=2\).

Step 2: Check x not equal to 0:
On each side of 0 the function is a constant (0 or 2), so it is continuous at every \(x\neq0\).

Step 3: Check x = 0:
Left limit \(=2\), right limit \(=0\). These are not equal, so the limit does not exist at 0, and \(f\) is discontinuous at \(0\) even though \(f(0)=2\).

Step 4: Conclusion:
The function is continuous for all \(x\) except \(x=0\). Option B.

Final Answer:
f is continuous everywhere except at 0. \[ \boxed{\text{(B) }\text{continuous for all }x\text{ except at }x=0} \]