Determine if f defined by

is a continuous function?


It is given that

We know that f is defined at all points of the real line.


Let k be a real number.


Case I: k ≠ 0,


Then f(k) =




Thus, f is continuous at all points x that is x ≠ 0.


Case II: k = 0


Then f(k) = f(0) = 0



We know that -1 ≤ ≤ 1, x ≠ 0


x2 ≤ 0




Similarly,



Therefore, f is continuous at x = 0.


Therefore, f has no point of discontinuity.


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