Maths · Limits, continuity and differentiability
Which one of the following is correct in respect of the function =\frac{\boldsym
Which one of the following is correct in respect of the function \( \boldsymbol{f}(\boldsymbol{x})=\frac{\boldsymbol{x}^{2}}{|\boldsymbol{x}|} \) for \( \boldsymbol{x} \neq \mathbf{0} \) and \( \boldsymbol{f}(\mathbf{0})=\mathbf{0} ? \)
- A. \( f(x) \) is discontinuous everywhere
- B. \( f(x) \) is continuous everywhere
- C. \( f(x) \) is continues at \( x=0 \) only
- D. \( f(x) \) is discontinuous at \( x=0 \) only
Step-by-step solution
For x ≠ 0, f(x) = x²/|x| = |x|, and f(0) = 0, so f(x) = |x| for all x. The absolute value function is continuous everywhere. Hence f(x) is continuous everywhere.
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