Question

Let $f(x+y)=f\left(x\right)+f\left(y\right)$ for all $x$ and $y$. If the function $f\left(x\right)$ is continuous at $x=0$, then $f\left(x\right)$ is continuous

• A. Only at $x=0$
• B. At $x\in R-\left\{0\right\}$
• C. For all $x$
• D. None of these

Answer 1

The correct option is D None of these

Given, $f(x+y)=f\left(x\right)+f\left(y\right)$; for all $x$ and $y$.

Since, $f\left(x\right)$ is continuous at $x=0$, we have $\underset{x\to 0}{lim}f\left(x\right)=f\left(0\right)$

To show that $f\left(x\right)$ is continuous at any point $a$, we shall prove that

$\underset{x\to a}{lim}f\left(x\right)=f\left(a\right)$

$\Rightarrow \underset{h\to 0}{lim}f(a+h)=f\left(a\right)$

Indeed, $\underset{h\to 0}{lim}f(a+h)=\underset{h\to 0}{lim}[f\left(a\right)+f\left(h\right)]$

$=f\left(a\right)+\underset{h\to 0}{lim}f\left(h\right)=f\left(a\right)+f\left(0\right)$

$=f(a+0)=f\left(a\right)$