Existence of Real-Valued Functions Satisfying Certain Properties.

(1)Is there a real-valued function f satisfies

the set {(x,f(x)),x belongs to R} is a second category subset of R2?

(2)Is there a real-valued function f satisfies

the set {(x,f(x)),x belongs to R} is non-measurable in the Lebesgue sense?

Re: a question about real-valued functions on R

Quote:

Originally Posted by

**mathabc** Is there a real-valued function f satisfies the set {(x,f(x)),x belongs to R} is non-measurable in the Lebesgue sense?

__Hint__: For any non Lebesgue measurable, choose the indicator function

Quote:

Is there a real-valued function f satisfies the set {(x,f(x)),x belongs to R} is a second category subset of R2?

__Hint__: Use the Baire's Category Theorem: *Every complete metric space is of second category*.

Re: a question about real-valued functions on R

... and is complete.

And with this, **we** end the lively conversation. :)

Re: a question about real-valued functions on R

Quote:

Originally Posted by

**FernandoRevilla** __Hint__: For any

non Lebesgue measurable, choose the indicator function

Just an observation: The graph of that function is a subset of a set of measure 0 (the line y=1).

Re: a question about real-valued functions on R