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If \( f(x)=\left|\log _{10} x\right| \), then at \( x=1 \)
(a) \( f(x) \) is continuous and \( f^{\prime}\left(1^{+}\right)=\log _{10} e \)
(b) \( f(x) \) is continuous and \( f^{\prime}\left(1^{+}\right)=-\log _{10} e \)
(c) \( f(x) \) is continuous and \( f^{\prime}\left(1^{-}\right)=\log _{10} e \)
(d) \( f(x) \) is continuous and \( f^{\prime}\left(1^{-}\right)=-\log _{10} e \)
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