calculus - A function $f:mathbb{R^+}→mathbb{R}$ satisfies the condition $f(ab)=f(a)+f(b)$ for $a,b>0$. prove that $f$ is continuous on $mathbb{R^+}$.

A function $f:\mathbb{R^+}→\mathbb{R}$ satisfies the condition $f(ab)=f(a)+f(b)$ for $a,b>0$. If $f(x)$ is continuous at $x=1$, then prove that $f$ is continuous on $\mathbb{R^+}$.

Attempt:

I have proved $f(1)=0$, $f(1/c)=-f(c),~~ c>0$.

Consider a sequence $\{x_n\}$ in $\mathbb{R^+}$ converges to $c>0$, then $$\lim_{n\to \infty}f(x_n)=\lim_{n\to \infty}f(\frac{x_n}{c}c)=\lim_{n\to \infty}f(\frac{x_n}{c})+\lim_{n\to \infty}f(c)$$

how to proceed further to use sequential criterion of continuity to prove that $f$ is continuous on $\mathbb{R^+}$.

My aim is to use sequential criterion of continuity $x_n\to c$ and $f(x_n)\to f(c)$ implies $f$ is continuous.

Please help.

Answer

Using your desired approach, we have by conditinuity of scalar multiplication that $x_n/c\to c/c=1$. So, as $f$ is continuous at $1$ by hypothesis, we have, as you wrote,
$$\lim_{n\to \infty}f(x_n)=\lim_{n\to \infty}f\left(\frac{x_n}{c}c\right)=\lim_{n\to \infty}f\left(\frac{x_n}{c}\right)+\lim_{n\to \infty}f(c)=f(1)+f(c)=f(c)$$ (as $f (1)=f(1\cdot 1)=f(1)+f(1)$ implies $f(1)=0$)
and so $f$ is continuous at $c$ as for any sequence $\{x_n\}$ converging to $c$, we have that $f(x_n)\to f(c)$.