If a function f: R → R satisfy the equation f ( x + y ) = f ( x ) +...

∵ \lim_{x \to 0}f \left(x\right) = f \left(0\right)$</p><p>Let$a$be any point</p><p>Now, at$x = a, \lim_{x \to a}f \left(x\right) = \lim_{h \to 0}f \left(a + h\right)$</p><p>$= \lim_{h \to 0}f \left(a\right) + \lim_{h \to 0}f \left(h\right)$</p><p>$= f \left(a\right) + f \left(0\right) = f \left(a + 0\right) = f\left(a\right)$</p><p>$∴ f \left(x\right)$is continuous at$x = a$, where$a$is any arbitrary point.</p><p>Hence,$f $\left(x\right)$ is continuous for all $x$.