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$$f(x) = \begin{cases} 1 & x \in [0,\frac 12] \\ 0 & x \in (\frac 12, 1] \end{cases}$$

The function is discontinuous at $\frac 12$. How do you modify the value of this function on set of zero measure ( $[0,\frac 12]$) such that we get continuous function on $[0,1]$?

I try to modify this function, but I cannot change it become continuous function. So I think it cannot modify the value on set of zero measure to become the continuous function. Could you help me? Thank you so much.

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It's not possible. –  user61527 Jan 1 at 21:39
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Why do you think this is possible? (It's not.) –  Potato Jan 1 at 21:39
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It is impossible. No matter how you change $f$ on a set of measure zero, it will still be discontinuous at $x={1 \over 2}$. –  copper.hat Jan 1 at 21:39
    
You can modify the function of a set of arbitrarily small measure and make it continuous. –  copper.hat Jan 1 at 21:40
    
Could you please show me more detail with the above function? –  user52523 Jan 1 at 21:51

1 Answer 1

up vote 4 down vote accepted

The goal isn't possible. Suppose that we could do this; since $f$ is $1$ almost everywhere on $[0, \frac 1 2]$, we can find a sequence $x_n \to \frac 1 2$ from below such that $f(x_n) = 1$ for all $n$. Likewise find a sequence $y_n \to \frac 1 2$ from above for which $f(y_n) = 0$ for every $n$. Hence the function is not continuous at $\frac 1 2$.

You can change the function on a set of arbitrarily small measure by inserting a little line segment connecting the points $(\frac 1 2 - \epsilon, 1)$ and $(\frac 1 2 + \epsilon, 0)$.

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could you explain more detail for me? Thank you so much. –  user52523 Jan 1 at 21:54
    
@user52523 Changing on a set of measure $0$ does very little to the function. There are points not changed on every interval contained in $[0, 1]$. –  user61527 Jan 1 at 22:00
    
thank you, and could you show me the way of modifying this function on set of arbitrarily small measure more detail? And could you give me some reference related. Thank you so much. –  user52523 Jan 1 at 22:12
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@user52523 Replace the function on the interval $(1/2 - \epsilon, 1/2 + \epsilon)$ with the line segment; you'll end up with a function which is piecewise linear. As far as related references, any book on basic measure theory would be good, such as Folland, Rudin or Royden. –  user61527 Jan 1 at 22:14
    
Thank you so much for your guide. –  user52523 Jan 1 at 22:19

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