Please see the links

http://www.madore.org/~david/math/padics.pdf

http://www.maths.gla.ac.uk/~ajb/dvi-ps/padicnotes.pdf

Could clarify the query a little? When you say representation, do you mean group homomorphism? And are you thinking of the p-adics (Qp or Zp?) with their additive or multiplicative group structure (Qp* or Zp*, if additive).

For continuous homomorphisms, these things are well-known, going between additive and multiplicative structures being failrly straightforward using exp or log on the p-adic or real side. For multiplicative homomorphisms, these things arise in the theory of Grossencharacters (the idelic formulation) . Any number of standard Number Theory texts that deal with Grossencharacters in detail, when the continuous (multiplicative) homomorphisms from  local fields to the complex numbers C have to be specified (ones into R* are easy special cases of these), can be referenced for this. Maybe they aren't the best references, but Lang's Algebraic Number Theory and relevant parts of the 1967 Algebraic Number Theory, edited by Cassels and Frohlich, (e.g. the Tate thesis chapter) are two that I am familiar with.

Basically, there aren't many homomorphisms, because, on the p-adic side, you have profinite totally disconnected, compact groups at the core, and on the real/complex side you have connected (or almost connected) groups with no small subgroup.

For example, continuous additive homomorphisms of Zp -> R: Zp is compact, so we get compact image under a continuous map. R, as an additive topological group, has no compact subgroup except the identity subgroup {0}. So, the only such homomorphism is the trivial one mapping all elements to 0.

For Qp -> R, the image has to factor through Qp/Zp by the above, which is a discrete p-torsion topological group. R (as an additive group) has no non-trivial torsion, so the Qp/Zp -> R homomorphism has to be trivial, and again you find that the only continuous additive homomorphism from Qp -> R is the trivial one.

Similar things occur if you consider multiplicative structures: Zp* = p-adic units  is a profinite group and Qp* = Zp* X Z, as topological groups, where Z is the additive group of integers with the discrete topology (the identification comes from choosing a uniformiser - usually p - which corresponds to the generator of the Z factor).

But you might mean representation in another sense, in which case this is all irrelevant :-)

1.As for the post of colleague R.C.Mittal, I guess that the topic starter needs not the basic facts about $p$-adic numbers the colleague addresses him to. I even guess that the topic starter is aware of advanced monographs on $p$-adic analysis like the classical Mahler's monograph "$p$-adic numbers and their functions" ore more recent books by Schikhof "Ultrametric calculus",

by Gouvea "Arithmetic of $p$-adic modular forms", by Gouvea "$p$-adic numbers", by Vladimirov et. al. "$p$-adic mathematical physics", several books of Khrennikov on applications of the $p$-adic theory to various disciplines, etc., etc. There are numerous books and numerous papers on the subject, but I guess that the topic starter needs something more definite. My two cents follows.

2. The most known continuous mapping of $p$-adic numbers into reals is the Monna map which is defined as follows: Represent a $p$-adic number $z$ in a canonical form (this representation is unique) $z=\sum_{i=-k}^\infty\alpha_ip^i$ where $k$ is a non-negative (real) integer, $\alpha_i\in\{0,1,\ldots,p-1\}$. The Monna map puts into the correspondence to $z$ the real number $mon(z)=\sum_{i=-k}^\infty\alpha_ip^{-i-1}$. It is clear that the most interesting part is the action of the Monna map on the $p$-adic integers which are represented as $\sum_{i=0}^\infty\alpha_ip^i$: Contrasting to real numbers, the "fractional part" of a $p$-adic number always has a finite canonical representation while the "integral part" not. So further i will speak only of mappings of the space of $p$-adic integers $\mathbb Z_p$ into reals. The space $\mathbb Z_p$ is a unit ball w.r.t. $p$-adic metric and is a sort of analog of a real unit interval. The Monna mapping $mon$ is a continuos mapping of $\mathbb Z_p$ into $[0,1]$ w.r.t the $p$-adic metric on $\mathbb Z_p$ and standard real metric on $[0,1]$. Moreover, $mon$ is a measure-preserving mapping w.r.t the probability Haar measure on $\mathbb Z_p$ and Lebesgue measure on $[0,1]$.

3. The  idea of the construction the Monna map is based on can be extended as follows: Take a real number $\beta>1$ such that the integral part of $\beta$ is $p-1$ and to every $p$-adic integer $z=\sum_{i=0}^\infty\alpha_ip^i$ put into the correspondence the real number $\sum_{i=0}^\infty\alpha_i\alpha_i\beta^{-i-1}$. In this case we deal with the so-called $\beta$-representations which were introduced by Parry and Renyi and now are rather popular research area, see. e.g. numerous papers of Frougny, Sakarovitch, Nikita Sidorov, and many others (too long to list all of them, sorry).

4. There are numerous generalizations for the case when $\beta$ is negative or even a complex number, and they are also related to real (or complex) representations of infinite words that correspond to $p$-adic numbers and have tight relations to automata theory, cf. e.g. the monograph by Lothaire "Algebraic combinatorics on words". The basic idea is that to a $p$-adic integer $z=\sum_{i=0}^\infty\alpha_ip^i$ there corresponds a (left-) infinite word $\ldots\alpha_2\alpha_1\alpha_0$ over a $p$-symbol alphabet; the word may be treated as a representation of a real number in some radix system (e.g., as a $\beta$-representation) and then one can study what reals (or sets of reals) obtained this way can be recognized by finite automata, or transformed by automata, etc, c.f., e.g., the monograph by Allouche and Shallitt "Automatic sequences". From this point of view, real representations of $p$-adic numbers give some information of behavior of automata and of the corresponding discrete dynamical system.

5. As automata map infinite words to infinite words, the automata mappings can be considered as mappings of $p$-adic integers to $p$-adic integers: Actually every automaton mapping is a continuous (w.r.t. the $p$-adic metric) function defined on the space of $p$-adic integers $\mathbb Z_p$ and valuated in $\mathbb Z_p$. Moreover, this mapping is 1-Lipschitz (=satisfies the Lipschtz condition with a constant 1) w.r.t. the $p$-adic metric and vice versa, every 1-Lipschitz mapping $\mathbb Z_p\to\mathbb Z_p$ can be performed by an automaton. From this view, real representations of $p$-adic numbers can visualize the automaton mapping (or the evolution of a discrete dynamical system). But being used for such a visualization, both the Monna map and $\beta$-representations can only give an information about short-time behavior of a system since they put the earliest inputted/outputted symbols to most significant positions in real representations. To study long-term behavior we need maps that act somehow in inverse direction. These "maps" are not maps in a strict sense of the word since put into the correspondence to a single $p$-adic integer some set (maybe, even infinite) of real numbers rather than a single number but these "mappings" can also be judged as 'continuous representations' in some meaning.

6. Namely, given a $p$-adic integer $z=\sum_{i=0}^\infty\alpha_ip^i$ we put into the correspondence to $z$ the set of all accumulation points of the sequence $p^{-k}\sum_{i=0}^{k-1}\alpha_ip^i$, $i=0,1,ldots$. From this view, the dynamics of an automaton map (or of a discrete system) can be represented as a real dynamics on the unit real square or, which is more natural, on the 2-dimensional torus. The representations of this can discover import ant peculiarities in discrete dynamics and are now under research, see e.g.

the monograph by Anashin (me) and Khrennikov "Applied algebraic dynamics" (deGryuter, 2009) and my papers on the theme, e.g. "The non-Archimedean theory of discrete systems" Math. Comp. Sci. 2012 (the paper can be found here on ReserchGate or in arxiv.org).

I am not sure that all these things are really that ones the topic starter is interested in but at least they are related to the question under the discussion. Sorry for the lengthy text, but I tried to be as concrete as possible.

Many thanks for answers! The question was about continuous additive group homomorphisms from Q_p to R.