Find all points of discontinuity of f, where f is defined by


The given function is

We know that if x > 0


|x| = -x and


x > 0


|x| = x


So, we can rewrite the given function as:



The function f is defined at all points of the real line.


Let k be the point on a real line.


Then, we have 3 cases i.e., k < 0, or k = 0 or k >0.


Now, Case I: k < 0


Then, f(k) = -1


= -1= f(k)


Thus,


Hence, f is continuous at all real number less than 0.


Case II: k = 0


= -1


= 1



Hence, f is not continuous at x = 0.


Case III: k > 0


Then, f(k) = 1


= 1 = f(k)


Thus,


Hence, f is continuous at all real number greater than 1.


Therefore, x = 0 is the only point of discontinuity of f.


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