Q. Let a function $f:R \rightarrow R$, where $R$ is the set of real numbers satisfying the equation $f(x + y) = f(x) + f(y),\ \forall\ x,\ y$ if $f(x)$ is continuous at $x = 0$, then
A) $f(x)$ is discontinuous, $\forall\ x \in R$
B) $f(x)$ is continuous, $\forall\ x \in R$
C) $f(x)$ is continuous for $x \in \{ 1,\ 2,\ 3,\ 4\}$
D) None of the above
Explanation
(b)
Let, $f(x) = x$
Which is continuous at $x = 0$
Also, $f(x + y) = f(x) + f(y)$
$\Rightarrow \ \ \ f(0 + 0) = f(0) + f(0)$
$= 0 + 0$
$\Rightarrow f(0) = 0$
$f(1 + 0) = f(1) + f(0)$
$\Rightarrow \ \ f(1) = 1 + 0$
$\Rightarrow \ \ f(1) = 1$
As, it satisfies it.
Hence, $f(x)$ is continous for every values of $x$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.