Bounded linear functional is necessarily continuous proof verification I want to prove that a bounded linear functional $f$, must be continuous.
I have defined: $f$ is bounded means that $\exists c> 0, |f(x)|\leq c\|x\|, \quad \forall x\in X$ and continuous means that $x_n\to x \implies f(x_n)\to f(x)$.
Proof: Let $f$ be bounded. Then $\exists c, |f(x)| \leq c\|x\|$
Then
\begin{align}
&\lim_{n\to\infty} |f(x_n-x)|\leq \lim_{n\to\infty} c\|x_n-x\|\\
\implies& \lim_{n\to\infty} |f(x_n)-f(x)|\leq c\|x-x\|=0\\
\implies& \lim_{n\to\infty} f(x_n)=f(x)
\end{align}
Is that all there is to it?
I used linearity in the second line, but what if we removed the condition that $f$ is linear. Is a non-linear bounded functional $f$ necessarily continuous

(1) Yes, that's all. 
(2) Without linearity, the statement becomes wrong, even in the $X = \mathbf R$ case. Consider for example, $f \colon \def\R{\mathbf R}\R\to \R$ given by 
$$ f(x) = \begin{cases} x & x \in [-1,1] \\ 0 & x \not\in [-1,1] \end{cases} $$
Then $f$ is bounded, as $|f(x)| \le |x|$ for all $x \in \R$, but not continuous.
