Why did we define the concept of continuity originally, and why it is defined the way it is? The concept of continuity is a very important idea in topology. Though I am using it all the time, but indeed I don't know what is the original purpose for us to define this concept. And I also don't understand why we define the continuity in topology as if $V$ is an open set in $Y$, $f^{-1}(V)$ is an open set in $X$ 
 A: The two main reasons we defined continuity, in my humble opinion, are as follows:


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*Continuity makes functions tame. Try to prove a proposition about arbitrary functions between topological spaces. You won't be able to get anywhere. Most topological spaces we deal with are so large that functions between them can behave in wild ways. If you don't forced them to respect some kind of structure coming from the topology, you won't be able to prove anything about them. If you force functions to ``respect topology'' by being continuous, then you'll get theorems such as: Compact sets map to compact sets under continuous maps. 

*In metric spaces continuity allows you to approximate. The standard $\epsilon\delta$-defintion of continuity allows you to approximate the value of a continuous function. It says: "If you give me a continuous function, a base point, and an error term, then I'll give you a range around your base point where your function is close to the value it takes at your base point." Approximation is good enough for most purposes, and it allows you to talk about limits. 
One way to rationalize the choice of definition of continuity in arbitrary topological spaces is that the definition we've chosen specializes to the $\epsilon\delta$-definition in metric spaces. There are some deeper reasons, but I think that that is a good place to start.
A: The idea of a point of closure expresses when a point is "close" to a set. And here I'm using "close" more loosely than in just a metric space distance. 
The idea of continuous maps is that they preserve points of closure. What I mean by that is that the image of a limit point is a point of closure of the image. So, continuous points don't "tear" points away from the sets they're close to.
Go ahead and check that the inverse image definition of continuity is equivalent to "the function preserves all points of closure."
A: Continuity was originally identified as an important property of functions of real variables, long before the invention of topology.  For example, the intermediate value theorem says that a function $f$ attains every value in an interval $[f(a), f(b)]$—but only if $f$ is continuous on $[a,b]$.  An important application of this is finding roots of equations that are too hard to solve symbolically.  If $f(a) < 0$ and $f(b) > 0$ we know there must be a root in $(a,b)$, and we can find it by subdividing the interval—but only if $f$ is continuous on $(a,b)$.
Similarly, as pgadey says elsewhere, continuous functions are those that can be approximated, which is crucial for obtaining approximate numerical solutions to problems of physics and engineering. 
Analysis was originally developed in order to study convergence and approximation properties of functions and sequences, questions like “which functions can be approximated by power series?” and “will my Fourier series sum to the correct function?”
Topology is an abstraction of ideas of continuity and convergence that already existed in analysis, created to study these ideas more carefully and in different settings.  General open sets are an abstraction of the properties of open intervals on the real line.  On the line, two sets are considered "distant" if they are contained within disjoint open intervals. In topology we replace this notion of distance with one involving open sets.  On the real line a function is continuous if it takes points that are "close together" in the domain to points that are "close together" in the codomain; points "far apart" in the codomain must be images of points "far apart" in the domain.  Topology reformulates the notion of closeness in terms of open sets: points that were in disjoint open sets ("sufficiently far apart") in the codomain must be in disjoint open sets ("sufficiently far apart") in the domain.  
Continuity is important in topology because it was already important in solving numerical problems in physics and engineering, and for this reason topology was invented to better understand continuity.
A: Besides the excellent answers already here, there's another way I think about it.
Given a topological space $(X, \mathcal T)$, there are actually two collections of subsets of $X$ which are important. Clearly $\mathcal T$ is such a collection, but so is the collection of closed sets.
The notation $(X,\mathcal T)$ is somewhat unfortunate, as it brushes the closed sets under the rug or off to the side even though they deserve equal footing with the open sets.
Now given two topological spaces $(X, \mathcal T_X)$ and $(Y,\mathcal T_Y)$, what functions $f:X\rightarrow Y$ should we consider to be the 'most important' or 'most fundamental'? I believe it should be those functions which preserve both the collection of open sets and the collection of closed sets.
It's tempting to consider open maps to be the most fundamental, but open maps need not send closed sets to closed sets. Dually, closed maps do not have to send open sets to open sets.
If we are putting closed sets on equal footing with open ones, another thing which hints that considering forward images is 'incorrect' is that intersections (the operation most important to closed sets) need not commute with forward images. But preimages do.
In fact, preimages commute with unions (the operation most important to open sets) and complements (the most important operation connecting open sets with closed sets) in addition. These observations hint that the 'most natural' morphism between topological spaces are the continuous ones instead of the open or closed (or open+closed) ones.
And comparing this definition with the more 'concrete' examples of topological spaces (such as metric spaces), we arrive at morphisms that we considered to be the most important in those contexts.
A: When a concept is defined, there are endless interpretations of that concept, some of them are extremely useful when working on Physics problems, but others are better when working on Geometry... The core of Topology resides in the concept of limit point of a set, by means of a empirical process of generalization we reach the nowadays well-know definition of topology. You should think of a continuous function to be one which trasform limit points of a set into limit points of the image of that set, this is the fact which tries the concept of continuity materialize...
A: Continuity has a history before topology. Ask yourself how to define a continuous function $\mathbb R \to \mathbb R$ - maybe one that you can draw the graph of without taking your pencil off the paper.
What makes it continuous? Well one candidate was the intermediate value property. Then people discovered pathological functions like $\sin \frac 1x$ near $x=0$ - or for a function which takes all real values in any interval (and in consequence has the intermediate value property) but is nowhere continuous try the extraordinary Conway base 13 function.
Then, in the metric context, epsilon-delta definitions were developed. But the thing about drawing the curve with a pencil got lost, because most continuous functions $\mathbb R \to \mathbb R$  have no defined arc length. Differentiable and smooth functions took over, since they were the ones people dealt with most often.
If you want to see another challenge to the formalisation of mathematics in this way, research the history of the Jordan Curve Theorem.
The idea of continuity developed into topology - the development of the two is intimately linked - one way of thinking about topology is to formalise it as what you get when continuity is the most significant concept you have. Now that is overstating it a bit, because most topological objects of interest have rather more structure than that. But that, to my way of thinking, is why topology and continuity go together. 
As for why continuity is defined as it is ... Topology deals with open sets. Continuity is concerned with functions. It just happens that the "inverse image of an open set under a function is open" coincides with the best intuitions we had of continuity before we abstracted it from a metric context.
A: The concept of continuity in topological space is crucial because of that if there exist a continuous function with the continuous inverse between two topological spaces, than in topological point of view they are identical. 
The main concept in topological space (which help us define that space) is the open set. Than dealing with 'equivalences' in terms of topology must be transmitted with open sets.
A: Continuity was introduced by Cauchy in 1821 in his famous textbook Cours d'Analyse. Here a function $f$ is continuous in a range if for all $x$ in the range, if $\alpha$ is infinitesimal then $f(x+\alpha)-f(x)$ is necessarily infinitesimal as well. This can be paraphrased in terms of real numbers alone (and was so paraphrased by Weierstrass and others 50 years later) but the resulting epsilon, delta definition does not fit as well with our intuitive perceptual notion of continuity as does Cauchy's original definition. The reformulation in terms of topologies is even further removed but is technically convenient. Of course Cauchy knew that $\frac{|x|}{x}$ is discontinuous though I am not sure about Fourier.
A: I think the answer is simple: continuity is a natural thing (maybe the natural thing) to study when the spaces you're dealing with all have well-defined open sets.
More than being the study of open sets, topology really is the study of continuous functions between spaces in the most general possible way. A set with a topology is the most general environment in which the concept of continuity can be discussed.
