A non-measurable function is one that cannot be assigned a definite value for every point in its domain. This typically occurs when the function is discontinuous or has an infinite number of discontinuities.
It is important to make this distinction because a function with non-measurable values may still be measurable, while a non-measurable function cannot be measured at all. This can have implications for the validity of mathematical proofs and theorems.
The absolute value of a non-measurable function may still be measurable, while the function itself is not. On the other hand, raising a non-measurable function to the power of 2 will also result in a non-measurable function.
Yes, a non-measurable function may still have measurable properties such as continuity, integrability, and differentiability. However, these properties may not hold for all points in the function's domain.
One example is the Dirichlet function, which takes on the value of 1 for rational numbers and 0 for irrational numbers. Another example is the Weierstrass function, which is continuous everywhere but differentiable nowhere.