Is there a Continuous Function f:R-->R Discontinuous at All Other Numbers?

nikolany
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Is there a function f:R-->R that is continuous at π and discontinuous at all other numbers?

Thx

 
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Consider the function f(x)=0 for x irrational and f(x)=x for x rational. Where is that continuous? Can you modify that for your purposes?
 
Dick said:
Consider the function f(x)=0 for x irrational and f(x)=x for x rational. Where is that continuous? Can you modify that for your purposes?

So f(x)=(x-π)^18 for x rational and 0 otherwise should work I think!

Thank you very much
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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