Discussing continuity of a function

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The function f defined on [0,1] takes the value x for rational x and x^2 for irrational x. To determine continuity, one must apply the definition of continuity at specific points, such as 0 and sqrt(2). The existence of both rational and irrational numbers in any interval is crucial for evaluating limits. Analyzing these limits will help establish whether the function is continuous at those points. Understanding these concepts is essential for solving the problem effectively.
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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.

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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.
 

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