Continuity of function - Proof verification
Solution 1:
Yes. Your proof is correct.
But a different approach, sequentially continuous iff $\epsilon , \delta $ continuous.
Let, $(x_n) $ be any sequence such that $(x_n) \to x $
Then, \begin{align}|f(x_n) -f(x) | & \le K|x_n -x| \to 0 \text { as } n\to \infty\end{align}
$\implies f(x_n) \to f(x) $
Hence, $f$ is continuous at any point $x\in \Bbb{R}$