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Till now I've not understood the differences between ( function ) , (mapping ) , (application) ,

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    So, what is your question? – 2017-01-11
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    I'd say they are synonyms. – 2017-01-11
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    A function is a method that maps of **every** element in the domain to **a single** element in the range. That is, a function is a restricted type of mapping. – 2017-01-11

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Let's start with function. A function from one set $R$ to another $D$ (where possibly $D$ is the same as $R$ is a binary relation between elements of $R$ and elements of $D$, such that for very element of $R$ the relation specifies exactly one element of $D$. This can be thought of as a set of pairs $r_i, d_i$ such that every $r \in R$ appears in exactly one of those pairs.

A mapping, or map, is almost synonymous with function, but in specific branches of mathematics, "map" is defined to be a function with some additional property. For example, in topology, a map is a continuous function $f$ (a function such that for any given open set $y\subset D$, the pre-image of Y which is $\{ x\in R : f(x) \in Y \}$ is an open set in $R$.

In addition, in computer science and occasionally mathematics, one might talk about a one-to-many map.

Application of a map or function is the act of going from $r \in R$ to $d \in D$ according to the rules of that map or function.

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    A continuous function does not need to send open sets to open sets. That's the definition of an open function. – 2017-01-11
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    Yes, @positron0802, you are right. Continuous does not imply open, and open does not imply continuous. I will change that part of the answer accordingly. – 2017-01-11