What is the meaning of real valued functions of real variable ? and how is it different from the real valued functions of a single real variable?
Help will be appreciated.
They are just functions $\displaystyle f:\mathbb R^n\to\mathbb R$. When $\displaystyle n=1$, $\displaystyle f$ is a real-valued function of a single real variable.
The set of all functions $\displaystyle f:\mathbb R^n\to\mathbb R$ forms an integral domain with addition and multiplication defined as follows: for all $\displaystyle \mathbf x\in\mathbb R^n$, $\displaystyle (f+g)(\mathbf x)\equiv f(\mathbf x)+g(\mathbf x)$ and $\displaystyle (fg)(\mathbf x)\equiv f(\mathbf x)g(\mathbf x)$.