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1answer
36 views

Prove if $E$ is a Lebesgue measurable set, there exists a continuous function $f$ differing from $\chi_{E}$ on a set of measure $< \epsilon$?

I am reviewing my analysis notes, and I don't really understand the proof given by my professor. He first proved if $E$ is a Lebesgue measurable set and $\epsilon > 0$, then there is an open set ...
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0answers
491 views

Continuity of a Characteristic function

Let $A$ be a subset of $\mathbb{R}^n$. Show that the characteristic function $\chi_A$ is continuous on the interior of $A$ and on $A^c$ but discontinuous on the boundary of $A$. My attempt: Suppose ...
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2answers
554 views

Evaluation of Derivative Using $\epsilon−\delta$ Definition

Consider the function $f \colon\mathbb R \to\mathbb R$ defined by $f(x)= \begin{cases} x^2\sin(1/x); & \text{if }x\ne 0, \\ 0 & \text{if }x=0. \end{cases}$ Use $\varepsilon$-$\delta$ ...
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0answers
161 views

Questions about characteristic functions and continuity

If there is a subset $S$ in $\mathbb{R}^n$ consider the characteristic function $\chi_S: \mathbb{R}^n \to \mathbb{R}$. i) what would be the value of $\chi_S(p)$ for any $p \in \mathbb{R}^n$? Attempt ...