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3 edited title; edited body

# $\lim_{x'\rightarrowx}f(x')\stackrel{?}{=}f(x)$foralmostall$x$if$f\inL^1(R^d)$?PointwiselimitatLebesgue'spoint

Dear MOs,

I am sorry if this problem is too elementary for someone. I just want to get confirmation. Suppose $f\in L^1(R^d)$. Since almost all points are Lebesgue points by The the Lebesgue differentiation theorem, can we say that for almost every $x\in R^d$,

$$\lim_{x'\rightarrow x} f(x') = f(x)\:?$$

I think it is true probably only for $d=1$. Does anyone know some results about this problem?

Thanks a lot!

Dear MOs,

I am sorry if this problem is too elementary for someone. I just want to get confirmation. Suppose $f\in L^1(R^d)$. Can Since almost all points are Lebesgue points by The Lebesgue differentiation theorem, can we say that for almost every $x\in R^d$,

$$\lim_{x'\rightarrow x} f(x') = f(x)\:?$$

I think it is true since almost all points are Lebesgue points by The Lebesgue differentiation theorem.probably only for $d=1$. Does anyone know some results about this problem?

Thanks a lot!

1

# $\lim_{x'\rightarrow x} f(x') \stackrel{?}{=} f(x)$ for almost all $x$ if $f\in L^1(R^d)$?

Dear MOs,

I am sorry if this problem is too elementary for someone. I just want to get confirmation. Suppose $f\in L^1(R^d)$. Can we say that for almost every $x\in R^d$,

$$\lim_{x'\rightarrow x} f(x') = f(x)\:?$$

I think it is true since almost all points are Lebesgue points by The Lebesgue differentiation theorem.

Thanks a lot!