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(C) To determine which function is continuous for all real values of $x$ and differentiable at $x = 0$:$1$. For $f(x) = |x|$,the function is continuou

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$\sin x$

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(C) To determine which function is continuous for all real values of $x$ and differentiable at $x = 0$:$1$. For $f(x) = |x|$,the function is continuous for all $x \in \mathbb{R}$,but it is not differentiable at $x = 0$ because the left-hand derivative is $-1$ and the right-hand derivative is $1$.$2$. For $f(x) = \log x$,the function is not defined for $x \le 0$,so it is not continuous for all real values of $x$.$3$. For $f(x) = \sin x$,the function is continuous for all $x \in \mathbb{R}$. Its derivative is $f'(x) = \cos x$. At $x = 0$,$f'(0) = \cos(0) = 1$,which is defined. Thus,it is differentiable at $x = 0$.$4$. For $f(x) = x^{\frac{1}{2}}$,the function is not defined for $x Therefore,the correct option is $C$.

The function which is continuous for all real values of $x$ and differentiable at $x = 0$ is

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