In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.

The directional derivative is a special case of the Gateaux derivative.

In mathematical analysis, and applications in geometry, applied mathematics, engineering, natural sciences, and economics, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or complex. However, the study of the complex valued functions may be easily reduced to the study of the real valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real valued functions will be considered in this article.

The domain of a function of n variables is the subset of ℝn for which the function is defined. As usual, the domain of a function of several real variables is supposed to contain an open subset of ℝn
n mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.

For example, a homogeneous real-valued function of two variables x and y is a real-valued function that satisfies the condition {\displaystyle f(\alpha x,\alpha y)=\alpha ^{k}f(x,y)}{\displaystyle f(\alpha x,\alpha y)=\alpha ^{k}f(x,y)} for some constant k and all real numbers α. The constant k is called the degree of homogeneity.
A study of limits and continuity in multivariable calculus yields many counterintuitive results not demonstrated by single-variable functions. there are scalar functions of two variables with points in their domain which give different limits when approached along different paths
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