Let f: R → R be defined by <span class="math-tex">\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {x\sin \left( {\frac{1}{x}} \right)}&{if\;x > 0}\\ 0&{x \le 0} \end{array}} \right.\)</span> Then
Continuity at 0: \(|x\sin\frac{1}{x}| \le |x| \to 0\), so the right limit is 0. The left limit is 0 and f(0) = 0. So f is continuous.
Differentiability at 0: RHD = \(\lim_{h \to 0^+}\frac{h\sin(1/h)}{h} = \lim\sin\frac{1}{h}\). This keeps oscillating, so the limit doesn't exist. So f is not differentiable at 0.
Answer: (C) continuous but not differentiable
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