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20 14 Fujian mathematical science
Your answer is wrong, your counterexample doesn't hold, your example is discontinuous at x=0, but according to the conditions, g(x) has a derivative, which means that the function g(x) is a continuous function.

If the condition says it is discontinuous at x=0, then your example is correct.

If the function is differentiable at x=0, then the function must be continuous at x=0.

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