A) \[f(x)\] is continuous for all x
B) \[f(x)\]is discontinuous for finite number of points
C) \[f(x)\]is discontinuous for finite number of points
D) None of the above
Correct Answer: B
Solution :
For\[x\ne 0\], \[f(x)=\left( 1-\frac{1}{1+x} \right)+\left( \frac{1}{1+x}-\frac{1}{1+2x} \right)\] \[+\left( \frac{1}{1+2x}-\frac{1}{1+3x} \right)\] \[+...+\left( \frac{1}{1+(n-1)x}-\frac{1}{1+nx} \right)\] \[=1-\frac{1}{1+nx}\] \[\Rightarrow \] \[f(x)=\underset{0\to \infty }{\mathop{\lim }}\,\left( 1-\frac{1}{1+nx} \right)=1-0=1\] and for\[x=0,\,\,f(0)=0\] \[\Rightarrow \] \[f(x)=\left\{ \begin{matrix} 1, & x\ne 0 \\ 0, & x=0 \\ \end{matrix} \right.\] Clearly, \[f(x)\] is discontinuous at\[x=0\].
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