Question

lf $f\left(xy\right)=f\left(x\right).f\left(y\right)\forall x,y\in R$. lf the function is continuous at one point $x=a$, then $f\left(x\right)$ is:

• A. continuous for all $x\in R-\left\{0\right\}$
• B. continuous forall $x\in R$
• C. discontinuous on $R$
• D. continuous at $x=0$

Answer 1

Answer: (A)

continuous for all $x\in R-\left\{0\right\}$

If$x=0andy=0$, $f\left(xy\right)=f\left(x\right)f\left(y\right)⟹f\left(0\right)=f(0{)}^2⟹f(0)=1andf(0)=0$ So $f\left(x\right)$ is not continuous at 0 but is continuous at all other points in the given domain.
Hence, our answer is "A"