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Let \( f(x) \) be a continuous function for all \( x \in R \) and \( f^{\prime}(0)=1 \). Then \( g(x)=f(|x|)-\sqrt{\frac{1-\cos 2 x}{2}} \) at \( x=0 \) is
(A) differentiable at \( x=0 \) and its value is 1
(B) differentiable at \( x=0 \) and its value is 0
(C) non-differentiable at \( x=0 \) as its graph has sharp turn at \( x=0 \)
(D) non-differentiable at \( x=0 \) as its graph has vertical tangent at \( x=0 \)
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