Question-30

Let \(\displaystyle f:\mathbb{R}\rightarrow \mathbb{R}\) such that \(\displaystyle f( x+y) =f( x) +f( y)\) for all \(\displaystyle x,y\in \mathbb{R}\). \(\displaystyle f\) is continuous at \(\displaystyle x=0\). Which of these statements is true if \(\displaystyle \mathbb{N,Z,Q}\) are the set of natural numbers, integers and rational numbers respectively?

• \(\displaystyle f\) is continuous everywhere
• \(\displaystyle f\) is continuous only when \(\displaystyle x\in \mathbb{N}\)
• \(\displaystyle f\) is continuous only when \(\displaystyle x\in \mathbb{Z}\)
• \(\displaystyle f\) is continuous only when \(\displaystyle x\in \mathbb{Q}\)

Answer

• \(\displaystyle f\) is continuous everywhere
• \(\displaystyle f\) is continuous only when \(\displaystyle x\in \mathbb{N}\)
• \(\displaystyle f\) is continuous only when \(\displaystyle x\in \mathbb{Z}\)
• \(\displaystyle f\) is continuous only when \(\displaystyle x\in \mathbb{Q}\)

Solution

Setting \(\displaystyle x=y=0\), we see that \(\displaystyle f( 0) =0\). Let us now compute the following limit for an arbitrary \(\displaystyle a\in \mathbb{R}\):

\[ \begin{equation*} \begin{aligned} \lim\limits _{h\rightarrow 0} \ f( a+h) & =\lim\limits _{h\rightarrow 0} \ \left(f( a) +f( h)\right)\\ & \\ & =f( a) +\lim\limits _{h\rightarrow 0} \ f( h)\\ & \\ & =f( a) +f( 0)\\ & \\ & =f( a) \end{aligned} \end{equation*} \]

We have used the continuity of \(\displaystyle f\) at \(\displaystyle x=0\). Since the limit at \(\displaystyle x=a\) is equal to the value of the function at \(\displaystyle x=a\), \(\displaystyle f\) is continuous at \(\displaystyle x=a\). As \(\displaystyle a\) was chosen to be an arbitrary point in \(\displaystyle \mathbb{R}\), \(\displaystyle f\) is continuous everywhere.