Upper semicontinuity of set-valued maps with open values Let $X$ and $Y$ be metric spaces. The $(\varepsilon,\delta)$-definition of continuity of single-valued maps can be rephrased as:

Let $f$ be a single-valued map from $X$ to $Y$. $f$ is continuous at $x_0 \in X$ if for every neighborhood $N_Y$ of $f(x_0)\in Y$, there exists a neighborhood $N_X$ of $x_0$ such that $f(N_X) \subset N_Y$.

In this book (Jean-Pierre Aubin and Hélène Frankowska, MR 2458436 Set-valued analysis, ISBN: 978-0-8176-4847-3, page 38.), this concept of continuity is extended to set-valued maps as:

Let $F$ be a set-valued map from $X$ to $Y$. $F$ is said to be upper semicontinuous at $x_0 \in X$ if for every neighborhood $N_Y$ of $F(x_0)\subset Y$, there exists a neighborhood $N_X$ of $x_0$ such that $F(N_X) = \bigcup_{x \in N_X}F(x) \subset N_Y$.

What bothers me is the possibility of $N_Y = F(x_0)$ when $F(x_0)$ is open. For example, if we let $X = Y = \mathbb{R}$, $F(x) = (0,x^2+1)$, and $x_0 \in \mathbb{R}$, then $F(x_0)$ itself is a neighborhood of $F(x_0)$ and $F(N_X) \not\subset F(x_0)$ for all neighborhood $N_X$ of $x_0$. Therefore, $F$ is not upper semicontinuous at $x_0$. However, it does not make sense for me; $F$ should be continuous, isn't it? Would you give me any reason why people do not put the additional condition like "$N_Y$ is a neighborhood of the closure of $F(x_0)$" on the definition?
 A: I think there is not much more to say than that most interesting results about upper semicontinuous set-valued maps involve closed-valued or even compact-valued maps. Indeed some authors choose to define upper semicontinutiy only for such maps. But there are good reasons not to. It helps to rephrase continuity notions for set-valued maps in terms of preimages of open sets, and here it turns out that there is more than one natural notion of a preimage.
Let $F:X\to 2^X$ be a set valued map and let $W\subseteq Y$. The upper inverse of $W$ under $F$ is
$$F^U(W)=\{x\in X:F(X)\subseteq W\}$$
and the lower inverse is 
$$F^L=\{x\in X:F(x)\cap W\neq\emptyset\}.$$
Then one can define upper semicontinuity by the requirement that the upper inverse of an open set is again open and lower semicontinuity by the requirement that the lower inverse of an open set is again open. If one identifies functions with set-valued maps whose values are singletons, both notions coincide with ordinary continuity there. Indeed, one often talks about hemicontinuity instead of semicontinuity nowadays exactly because a, say, upper semicontinuous function is in general not upper semicontinuous as a set-valued map.
Now these two notions of continuity for set-valued maps represents different aspects of continuity. The example you gave does satisfy lower hemicontinuity so it does satisfy an idea of continuity, just not the one you used. 
A very accessible reference for results on set-valued maps is Chapter 17 of Infinite Dimensional Analysis by Aliprantis and Border.
A: This map is single valued map. $f$ is upper semicontinuous  at $x_0$ if for every positive epsilon there exist a nbhd $U$ of $x_0$ such that $f(x) \le f(x_0)+\varepsilon$.
