Is f ( x ) = sin ⁡ ( sign ⁡ x e x )...

Ellen Chang

Ellen Chang

Answered

2022-07-04

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?

Answer & Explanation

kawiarkahh

kawiarkahh

Expert

2022-07-05Added 15 answers

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't affect the sign of x at all, so sign x e x = sign x
Passing 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?

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