Is f ( x ) = sin ⁡ ( sign ⁡ x e x ) a measurable function ? I remember a theorem that if f is a measurable function and g is continuous then g ∘ f is measurable. Now we know that the sin function is continuous but how can I prove that sign ⁡ x e x is measurable?

Question

Is   f  (  x  )  =  sin  ⁡  (  sign  ⁡  x      e    x    ) a measurable function ? I remember a theorem that if   f is a measurable function and   g is continuous then   g  ∘  f is measurable. Now we know that the sin function is continuous but how can I prove that   sign  ⁡  x      e    x   is measurable?

kawiarkahh · Accepted Answer

Note that the sign function outputs 1 for positive numbers, 0 for 0, and -1 for negative numbers. As       e    x   is always positive, it doesn&#039;t affect the sign of   x at all, so   sign  ⁡  x      e    x    =  sign  ⁡  xPassing this into sin just gets you a piecewise function with 3 outputs,   sin  ⁡  (  −  1  )  ,  0  ,  a  n  d  sin  ⁡  (  1  ). This is the sum of simple functions. What does that tell you?