and continuous everywhere, and is differentiable everywhere, except for zero. The cubic root is defined and continuous everywhere, and is differentiable everywhere, except for zero. Many common functions are not defined everywhere, but are continuous and differentiable everywhere where they are defined. For example: A rational function is a quotient of two polynomial functions, and is not defined at the zeros of the denominator. The tangent function is not defined for where is any integer. The logarithm function is defined only for positive values of the variable. Some functions are continuous in their whole domain, and not differentiable at some points. This is the case of: The square root is defined only for nonnegative values of the variable, and not differentiable at 0 (it is differentiable for all positive values of the variable). General definition A real-valued function of a real variable is a function that takes as input a real number, commonly represented by the variable x, for producing another real number, the value of the function, commonly denoted f(x). For simplicity, in this article a real-valued function of a real variable will be simply called a function. To avoid any ambiguity, the other types of functions that may occur will be explicitly specified. Some functions are defined for all real values of the variables (one says that they are everywhere defined), but some other functions are defined only if the value of the variable is taken in a subset X of ℝ, the domain of the function, which is always supposed to contain an interval of positive length. In other words, a real-valued function of a real variable is a function such that its domain X is a subset of ℝ that contains an interval of positive length. A simple example of a function in one variable could be: which is the square root of x. Image The image of a function is the set of all values of when the variable x runs in the whole domain of . For a continuous (see below for a definition) real-valued function with a connected domain, the image is either an interval or a single value. In the latter case, the function is a constant function. The preimage of a given real number y is the set of the solutions of the equation . Domain The domain of a function of several real variables is a subset of ℝ that