Yes, the Cauchy-Riemann condition is true for all analytic real valued functions. In which case they can be used to prove that the class of analytic real valued functions consists of the real constants.

But the Cauchy-Riemann conditions does not hold for arbitrary functions of two variables, regardless of they are real-, imaginary-, or complex-valued. 

it has to be pointed out that analytic functions ( complex variable) are guranteed just  with the existence of the first- derivative and this follows from Cauchy- Riemann conditions. The functions of two real variables do not fulfill this: the existence of the first derivatives  do not guarantee at all the existence of the succesive ones. So,  my opinion is to give a negative answer to your question.

Yes, the Cauchy-Riemann condition is true for all analytic real valued functions. In which case they can be used to prove that the class of analytic real valued functions consists of the real constants.

But the Cauchy-Riemann conditions does not hold for arbitrary functions of two variables, regardless of they are real-, imaginary-, or complex-valued. 

Dear Pooyan Goodarzi  

When the conditions of Cauchy - Riemann function of a complex variable f (x, y) = u (x, y) + iv (x, y), where (x, y) in R x R (or in some area of R x R), or z = x + i y in C (or in some area of C), which accepts only real values, i. e.. v (x, y) = 0 for every (x, y) in R x R (or in some area of R  x R), or for every z = x + i y in C (or in some area of C), are implemented, then this function is a real constant.  The opposite is also obviously true.Your sincerely, Anna Tomova.

The analogue of the Cauchy-Riemann equations for real C^(1) - functions of two real variables might be the Green formula. From this formula, which has a simple proof, the Cauchy theorem and then the Cauchy integral formula follow, via Cauchy - Riemann equations. The latter equations have also a simple proof for C-derivable functions. A consequence of the Green formula is: if (Drond)(Q)/Drond)x=(Drond)(P)/(Drond)y everywhere in K, then the integral on the boundary of K from Pdx + Qdy vanishes. This is a "real analogue" of the Cauchy theorem for C - derivable functions.

Cauchy-Riemann equations for real holds only for particular case i.e. (0,0). 

Let first explain which is the difference between the function f(z)=u(x,y)+iv(x,y) which seems to be a function of two real variables from one side. And from another side the function g(x,y)=w. From a general point of view both functions (we can say) are functions of a two dimentional vector and that's the common for them. Look now on their values: the values of the first are two dimentional vectors and the values of the second are real numbers (we can say - one dimantional vectors). More over

u(x,y) and v(x,y) in this case are elliptic functions.

So than what about g(x,y) - there is no analogue.

My final opinion: we cannot speak for Cauchy-Riemann equations in the case of real valued functions.

Why not, if u(x'y) = const in R and v(x, y) = 0 and f(z) = f(x + iy) = u(x,y) + i0 = const in R?

The question of the author is not clear, what is the domain of the function. Therefore, we will examine two cases:

1) The function is a real function of two real variables. She is an image of R x R to R and in this case we can not talk about conditions Cauchy - Riemann, which relate to the function of a complex variable;

2) If the function is a holomorphic function of a complex variable and taking real values, then it is a special case of function of a complex variable. We know that if  the real part is constant (in some area of C) then the function itself is also constant and the conditions of Cauchy - Riemann are implemented to this function: f(z) = f(x+ iy) = u(x, y) + i 0= (const, 0) .