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



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> 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 $f(N\_X) \subset N\_Y$.
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In this [book](https://books.google.com/books?id=cVUl3smcGlcC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false) (*Jean-Pierre Aubin and Hélène Frankowska*, MR 2458436 [**Set-valued analysis**](http://dx.doi.org/10.1007/978-0-8176-4848-0), ISBN: 978-0-8176-4847-3, [page 38](https://books.google.com/books?id=cVUl3smcGlcC&pg=PA38).), this concept of continuity is extended to set-valued maps as:



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> Let $F$ be a set-valued map from $X$ to $Y$. $F$ is said to be [upper semicontinuous](https://en.wikipedia.org/wiki/Hemicontinuity) 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 $F(N\_X) = \bigcup\_{x \in N\_X}F(x) \subset N\_Y$.
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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?



Upper semicontinuity of set-valued maps with open values
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 its *lower inverse* is


$$F^L(W)=\{x\in X:F(x)\cap W\neq\emptyset\}.$$
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](http://www.springer.com/gp/book/9783540295860) by Aliprantis and Border.


