If \( f(x+y)=f(x)+f(y) \) for all \( x,y\in R \) (set of real numbers) and \( f(x) \) is continuous at \( x=0 \), then identify the correct statement ?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( f(x) \) is continuous for all \( x \)
Putting x = y = 0 gives f(0) = 2f(0), so f(0) = 0. Continuity at 0 then means \( \lim_{h\to 0}f(h)=0 \).
Now let a be any real number. Since f(a + h) = f(a) + f(h),
\( \displaystyle \lim_{h\to 0}f(a+h)=\lim_{h\to 0}\left[f(a)+f(h)\right] \)
\( \displaystyle =f(a)+0 \)
\( \displaystyle =f(a) \)
So f is continuous at every real a.