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No, a measurable function may have a finite number of discontinuities (for the Riemann measure), or a countably infinite number of discontinuities (for the Lebesgue measure). It should also be bounded (have some upper and lower bound, or limit, in the domain that is being measured), to be measureable. At least, some unbounded functions are not measurable.

No, a measurable function may have a finite number of discontinuities (for the Riemann measure), or a countably infinite number of discontinuities (for the Lebesgue measure). It should also be bounded (have some upper and lower bound, or limit, in the domain that is being measured), to be measureable. At least, some unbounded functions are not measurable.

No, a measurable function may have a finite number of discontinuities (for the Riemann measure), or a countably infinite number of discontinuities (for the Lebesgue measure). It should also be bounded (have some upper and lower bound, or limit, in the domain that is being measured), to be measureable. At least, some unbounded functions are not measurable.

No, a measurable function may have a finite number of discontinuities (for the Riemann measure), or a countably infinite number of discontinuities (for the Lebesgue measure). It should also be bounded (have some upper and lower bound, or limit, in the domain that is being measured), to be measureable. At least, some unbounded functions are not measurable.

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No, a measurable function may have a finite number of discontinuities (for the Riemann measure), or a countably infinite number of discontinuities (for the Lebesgue measure). It should also be bounded (have some upper and lower bound, or limit, in the domain that is being measured), to be measureable. At least, some unbounded functions are not measurable.

Q: Is every measurable functions continuous

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yes, every continuous function is integrable.

Yes, a corner is continuous, as long as you don't have to lift your pencil up then it is a continuous function. Continuous functions just have no breaks, gaps, or holes.

Yes. For every measurable function, f there's a sequence of simple functions Fn that converge to f m-a.e (wich means for each e>0, there's X' such that Fn|x' -->f|x' and m(X\X')<e).

They are both continuous, symmetric distribution functions.

Measurable/ Make your goal able to beย Measured.

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yes, every continuous function is integrable.

Krzysztof Ciesielski has written: 'I-density continuous functions' -- subject(s): Baire classes, Continuous Functions, Functions, Continuous, Semigroups

All differentiable functions need be continuous at least.

Not at all.Y = x2 is a continuous function.

No. Not all functions are continuous. For example, the function f(x) = 1/x is undefined at x = 0.

Yes, all polynomial functions are continuous.

No. There are many common functions which are not discrete but the are not continuous everywhere. For example, 1/x is not continuous at x = 0 (it is not even defined there. Then there are curves with step jumps.

Jean Schmets has written: 'Spaces of vector-valued continuous functions' -- subject(s): Continuous Functions, Locally convex spaces, Vector valued functions

Frederick Bagemihl has written: 'Meier points and horocyclic Meier points of continuous functions' -- subject(s): Continuous Functions

Yes, a corner is continuous, as long as you don't have to lift your pencil up then it is a continuous function. Continuous functions just have no breaks, gaps, or holes.

An infinite sum of continuous functions does not have to be continuous. For example, you should be able to construct a Fourier series that converges to a discontinuous function.

Yes. For every measurable function, f there's a sequence of simple functions Fn that converge to f m-a.e (wich means for each e>0, there's X' such that Fn|x' -->f|x' and m(X\X')<e).