Discussing continuity of a function

kmeado07
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Homework Statement



Discuss the continuity of the function f defined for all x belongs to [0,1] by f(x)=x if x is rational and f(x)=x^2 is x is irrational.

Homework Equations





The Attempt at a Solution



I have no idea how to begin this question...some help would be great thanks!
 
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Maybe you should try checking some points. Is f continuous at 0? How about sqrt(2)?
 
You will need to know
1) The definition of "continuous at a point".
2) The fact that there exist rational numbers in any interval, no matter how small.
3) The fact theat there exist irrational numbers in any interval, no matter how small.
The last two should help you find the limit, or determine if it does not exist, at any point.
 

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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