JEE Main & Advanced Mathematics Limits, Continuity and Differentiability Question Bank Self Evaluation Test - Continuity and Differentiability

If \[f(xy)=f(x).f(y)\] for all x, y and f(x) is continuous at x = 2, then f(x) is not necessarily continuous in:

A) \[(-\infty ,\infty )\]

B) \[(0,\infty )\]

C) \[(-\infty ,0)\]

D) \[(2,\infty )\]

Correct Answer: A

Solution :

[a] Given, \[f(xy)=f(x).f(y)\] for all x, y, ?.(1)

\[f(x)\] is continuous at x = 2,

i.e.,\[\underset{x\to 2}{\mathop{Lt}}\,\,\,f(x)=f(2)...(2)\]

Let \[a\ne 0\]

Now, \[\underset{x\to a}{\mathop{Lt}}\,\,\,f(x)=\underset{h\to 2}{\mathop{Lt}}\,\,f\left( \frac{ah}{2} \right)\]

\[\left[ putting\,\,x=\frac{ah}{2}so\,\,that\,\,h=\frac{2x}{a} \right]\]

\[=f\left( \frac{a}{2} \right)\underset{h\to 2}{\mathop{Lt}}\,f(h)=f\left( \frac{a}{2} \right).f(2)=f\left( \frac{a}{2}.2 \right)=f(a)\]

Hence, \[f(x)\] is necessarily continuous at x = a for all \[a\ne 0\].

At x = 0, f(x) may or may not be continuous

Hence f(x) is not necessarily continuous in\[(-\infty ,+\infty )\].