Question

If \( f(x) = \frac{|x|}{x} \) for \( x \neq 0 \) and \( f(x) = 1 \) for \( x = 0 \), then the function is

• A. continuous but not differentiable at \( x = 0 \)
• B. differentiable but not continuous at \( x = 0 \)
• C. neither continuous nor differentiable at \( x = 0 \)
• D. continuous and differentiable at \( x = 0 \)

Hint:

For a function to be differentiable at a point, it must first be continuous at that point.

Answer 1

The Correct Option is C

Step 1: Check continuity at \( x = 0 \).
For continuity at \( x = 0 \), we check the limit of \( f(x) \) as \( x \to 0 \). We find that \( \lim_{x \to 0^-} f(x) = -1 \) and \( \lim_{x \to 0^+} f(x) = 1 \), so the function is not continuous at \( x = 0 \).

Step 2: Check differentiability at \( x = 0 \).
Since the function is not continuous at \( x = 0 \), it cannot be differentiable there.

Step 3: Conclusion.
Thus, the function is neither continuous nor differentiable at \( x = 0 \), corresponding to option (C).