Example of a continuous function which is not smooth 
I need a function which is continuous but not smooth ( not a $C^{\infty})$. 

Smooth functions are those  whose derivatives of all order exists. For example $f(x)= e^{x}$ is a smooth function  while $f(x)=|x|$ is not smooth as derivative at $0$ does not exist.
But what I require is functions in from $\mathbb{R}^{n} $ to $\mathbb{R}^{m}$. 
For simplicity it is enough to give functions from $\mathbb{R}^{2} $ to $\mathbb{R}^{2}$. I have examples of discontinuous functions from $\mathbb{R}^{2} $ to $\mathbb{R}^{2}$ , like $\frac{xy}{x^{2}+y^{2}}$  which is not continuous at $(0,0)$.
 A: Your example $x\mapsto|x|$, mapping $\mathbb R$ to $\mathbb R$ continuously but not smoothly, can be trivially upgraded to map $\mathbb R^2$ to $\mathbb R^2$ continuously but not smoothly: $(x,y)\mapsto(|x|,y)$.  ($(x,y)\mapsto(|x|,|y|)$ works too.) The same ideas can be used for higher-dimensional spaces.
A: Such functions are commonly used in approximations, such as by piecewise polynomials/splines.  For example we might "triangulate" a region of the plane and create piecewise linear functions on the triangles that globally form a continuous  but not differentiable function.  Linear interpolation between corners of triangles will accomplish this.
This readily generalizes to maps $f:\mathbb{R}^n \to \mathbb{R}$ by subdividing a region of $n$-space into conforming simplices, i.e. that meet face to face.  Once again an arbitrary assignment of values at the collected vertices extends to a continuous (but typically not smooth) function on the whole region by linear interpolation, often implemented via barycentric coordinates.
Obtaining vector-valued functions $f:\mathbb{R}^n \to \mathbb{R}^m$ in this way is easy; we have no particular requirements that separate "output" coordinates should be related in any way (as far as this Question is concerned).
Obtaining higher-order continuity, e.g. continuity of the first- and second-order derivatives, is a bigger challenge.  The need for such things often comes up in the context of finite element approximations.
One of the easier ways to obtain interpolations that have continuous first derivatives is with bicubic interpolation on a regular (rectangular) grid.
A: Consider the function $f:\Bbb R^2\to \Bbb R^2$ given by
$f(x,y)=(|x-y|,|x+y|)=(f_1(x,y),f_2(x,y))$.
$f$ is continuous since both $f_1,f_2$ are so.
Now $f$ is not differentiable since $f_1(x,y)=|x-y|$ is not differentiable at $(0,0)$.
The partial derivative of $f_1$ at $(0,0)$ :
$\lim_{h\to 0}\dfrac{f_1(h,0)-f_1(0,0)}{h}=\dfrac{|h|}{h}$ does not exist.
