How to find the points that the function is not analytic?

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The discussion focuses on identifying points where a function is not analytic, specifically using the rational function f(x)=(x+2)/((x-1)*(x-2)). It is established that a function is not analytic at points where the denominator equals zero, which in this case occurs at x=1 and x=2. However, participants agree that there is no universal method for determining non-analytic points across all functions, emphasizing the need for specific analysis based on the function's characteristics.

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For example,

f(x)=(x+2)/((x-1)*(x-2))

We know that f(x) is not analytic for x=1 and x=2 (from our eye) ,
but do there exist a general method to find out these points?

Thank you
 
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For rational functions, of course, the function is not analytic if and only if the denominator is 0 (where the function is not defined) but there is no general way of determining where a function is not analytic.
 

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