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Let $f(x)$ be a continuous function on an interval $X$, and let $\operatorname{supp}\left(f\right)=X$. Let, also: $$g(x):=\int_{x-\frac12}^{x+\frac12} f(t)dt\, .$$ What should be the support $X$ to have $\operatorname{supp}\left(g\right)=\left(-\frac12,\frac12\right)$?

My attempt. Let $f=1$ on $X=[a,b]$ with $b>a$ and $a,b\in\mathbb R$. We have only the following nonempty cases:

1) $aa+\frac{1}{2}$; thus: $x\in I_1=\left(a+\frac{1}{2},b-\frac{1}{2}\right)$. 2) $a

Then $X=I_1 \cup I_2 \cup I_3=\left(a-\frac{1}{2}, b+\frac{1}{2}\right)$ which is included in $\left(-\frac12,\frac12\right)$ only if $a>0$ and $b<0$. But this is a contradiction.

We obtain the same conclusion starting from $b-\frac{1}{2}

So... $X=\emptyset$?

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    My attempts give $X=\emptyset$, but I'm not sure... – 2017-02-05
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    What type of attempts? Please explain what you have tried so far. – 2017-02-05
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    @πr8, thank you, I've edited my post. – 2017-02-05
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    Looking at some simple examples would instantly destroy 90% of this post, sorry to say. – 2017-03-10

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