Limits Continuity And Differentiability Question 109
Question: 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:
Options:
A) $ (-\infty ,\infty ) $
B) $ (0,\infty ) $
C) $ (-\infty ,0) $
D) $ (2,\infty ) $
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Answer:
Correct Answer: A
Solution:
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( \frac{ah}{2} ) $
$ [ puttingx=\frac{ah}{2}sothath=\frac{2x}{a} ] $
$ =f( \frac{a}{2} )\underset{h\to 2}{\mathop{Lt}}f(h)=f( \frac{a}{2} ).f(2)=f( \frac{a}{2}.2 )=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 ) $ .
BETA
