The function f(x) = {[1,|x| ≥1][1/n^2,1/n<|x|<1/(n-1),n=2,3...][0,x=0] A. is discontinuous at finitely many points B. is continuous everywhere

Question

The function \(f(x) = \begin{cases} 1 &, \quad |x| ≥{1}\\ \frac{1}{n^2} &, \quad \frac{1}{n}< |x|<\frac{1}{n-1},n=2,3,...\\0&,\quad x=0 \end{cases} \) 

A. is discontinuous at finitely many points 

B. is continuous everywhere 

C. is discontinuous only at x = ± \(\frac{1}{n}\), n ∈ Z - {0} and x = 0 

D. none of these

Answer 1

Option : (C)

(i) A function f(x) is said to be continuous at a point x = a of its domain, if 

 \(\lim\limits_{x \to a}f(x)\) = f(a)

 \(\lim\limits_{x \to a^+}f(a+h)\) = \(\lim\limits_{x \to a^-}f(a-h)\) = f(a)

Given :-

\(f(x) = \begin{cases} 1 &, \quad |x| ≥{1}\\ \frac{1}{n^2} &, \quad \frac{1}{n}< |x|<\frac{1}{n-1},n=2,3,...\\0&,\quad x=0 \end{cases} \) 

Therefore,

f(x) is discontinuous only at  x = ± \(\frac{1}{n}\), n ∈ Z - {0} and x = 0