Examples of presheaves which are not sheaves 1) Consideremos $\mathbb{C}$ con la topología usual. Definimos el prehaz de las $\textbf{funciones acotadas}$ $\mathcal{F}:\textbf{Top}(X)\to \textbf{Ab}$ de la siguiente manera:
$$\mathcal{F}(U)=\{f:U\to \mathbb{C} \mid f\text{ acotada}\}.$$
Este prehaz no es un haz.
2)  Consideremos $\mathbb{C}$ con la topología usual. Definimos el prehaz de las $\textbf{funciones acotadas}$ $\mathcal{F}:\textbf{Top}(X)\to \textbf{Ab}$ de la siguiente manera:
$$\mathcal{F}(U)=\{f:U\to \mathbb{C} \mid f\text{ holomorfa y acotada}\}.$$
Este prehaz no es un haz.
Más ejemplos de prehaces que no sean haces?
Muchas gracias
Translation:
1): Consider $\mathbb{C}$ with the usual topology. Define the presheaf of bounded functions $\mathcal{F}:\textbf{Top}(X)\to \textbf{Ab}$ as follows:
$\mathcal{F}(U)= \lbrace f:U\to \mathbb{C} \mid f \text{ is bounded} \rbrace$
This presheaf is not a sheaf.
2): Consider $\mathbb{C}$ with the usual topology. Define the presheaf of bounded functions $\mathcal{F}:\textbf{Top}(X)\to \textbf{Ab}$ as follows:
$\mathcal{F}(U)= \lbrace f:U\to \mathbb{C} \mid f \text{ is bounded and holomorphic} \rbrace$
This presheaf is not a sheaf.
More examples of presheaves which are not sheaves?
Thanks a lot.
 A: Let me add one way to create some examples:
If $\mathcal{F}$ is a sheaf (of for example abelian groups, rings, etc.), then $\mathcal{F}(\emptyset)$ will be the terminal object in the category where the sheaf takes its values. That means you can find examples of presheaves which are not sheaves by  preventing $\mathcal{F}(\emptyset)$ to be the terminal object. A very prominent example for that is given by constant presheaves:
Take for example some non-trivial abelian group $G$, and define a presheaf of abelian groups by $\mathcal{F}(U) = G$ (the restriction maps being the identity $\text{id}_G$)
In general this example not only fails to be a sheaf because of the above reason, but also because gluability tends to fail.
More examples arise in the following setting:
If you have a morphism $\varphi \colon \mathcal{F} \rightarrow \mathcal{G}$ of for example sheaves of abelian groups, then the presheaves $\text{im}(\varphi)$ and $\text{coker}(\varphi)$ defined by $\text{im}(\varphi)(U) = \text{im}(\varphi_U)$ and $\text{coker}(\varphi)(U) = \text{coker}(\varphi_U)$ will not be sheaves in general, where once again gluability tends to be the problem.
You can for example take a look at the topological space $X = \mathbb{C}$ you were considering yourself and the morphism $\text{exp} \colon \mathcal{O}_X \rightarrow \mathcal{O}^{\times}_X,$ where $\mathcal{O}_X$ is the sheaf of holomorphic functions and $\text{exp}$ is given by locally taking the exponential.
More examples arise from other constructions. When you try to mimic a construction known from for example abelian groups, then it will not necessarily respect the sheaf axiom and just be a presheaf at first. 
That also shows why we will need to work with sheafification in general. 
