Linear and nonlinear operator on normed space and its properties My first question is :
We know every bounded linear operator is continuous operator , and every continuous linear operator is bounded  operator , in other words the continuousness and boundedness are equivalent in linear operator . But if the operator is nonlinear can I find bounded operator but it isn't continuous , and can I find continuous operator but it isn't bounded . 
My second question is : 
If $T$ is bounded linear operator then the zero space $N(T)$ is closed ,$N(T)$ is space of all $x \in X $ such that $Tx=0$ , but they say $R(T)$ the range of the operator isn't necessary to be closed how can I show that I mean an example . 
And if the operator $T$ is nonlinear  is it necessary $N(T)$ to be closed ???
 A: (1) Define $F \colon c_0 \to \def\K{\mathbf K}\K$ by 
$$ F(x) = \sum_{n=0}^\infty x_n^n $$
Then $F$ is continuous, as the series converges locally uniform, but $F$ is unbounded, as the elements $x^{(n)} = (1, \ldots, 1,0, \ldots) \in \bar B_{c_0}$ have 
$$ \def\norm#1{\left\|#1\right\|} \norm{x^{(n)}} = 1,\qquad \def\abs#1{\left|#1\right|}\abs{F(x^{(n)})} = n, $$
$F(\bar B_{c_0})$ is unbounded.
(2) If $X$ is any normed space $\ne 0$, the map 
$$ G \colon x \mapsto \begin{cases} \frac x{\norm x} & x \ne 0\\ 0 & x = 0 \end{cases} $$
is discontinuous, as for $x \ne 0$, $\norm{G(x)} = 1 \not \to 0$ for $x \to 0$, but bounded, as its image is contained in $\bar B_X$, an hence $G$ maps bounded sets to bounded sets.
(3) Define $T \colon \ell^1 \to c_0$ by $Tx = x$. Then $T$ is continuous, as 
$$ \norm{Tx}_\infty = \sup_n \abs{x_n} \le \sum_{k=0}^\infty \abs{x_n} = \norm x_1 $$
but the images is not closed, as 
$$x^{(n)} = \left(1, \frac 12, \ldots, \frac 1{n+1}, 0, \ldots \right) \in \ell^1, $$
but its $c_0$-limit 
$$ x = \left(1, \frac12, \ldots, \right) $$
does not lie in $\ell^1 = T\ell^1$.
(4) For any continuos map, inverse images of closed sets are closed. As $\{0\}$ is closed in a normed space, $N(T) = T^{-1}[\{0\}]$ is closed, for any continuous map.
