chosen small enough for having the image by f of the interval of radius centered at contained in the interval of length centered at A function is continuous if it is continuous at every point of its domain. The limit of a real-valued function of a real variable is as follows. Let a be a point in topological closure of the domain X of the function f. The function, f has a limit L when x tends toward a, denoted if the following condition is satisfied: For every positive real number ε > 0, there is a positive real number δ > 0 such that for all x in the domain such that If the limit exists, it is unique. If a is in the interior of the domain, the limit exists if and only if the function is continuous at a. In this case, we have When a is in the boundary of the domain of f, and if f has a limit at a, the latter formula allows to "extend by continuity" the domain of f to a. Calculus One can collect a number of functions each of a real variable, say into a vector parametrized by x: The derivative of the vector y is the vector derivatives of fi(x) for i = 1, 2, ..., n: One can also perform line integrals along a space curve parametrized by x, with position vector r = r(x), by integrating with respect to the variable x: where · is the dot product, and x = a and x = b are the start and endpoints of the curve. Theorems With the definitions of integration and derivatives, key theorems can be formulated, including the fundamental theorem of calculus, integration by parts, and Taylor's theorem. Evaluating a mixture of integrals and derivatives can be done by using theorem differentiation under the integral sign. Implicit functions A real-valued implicit function of a real variable is not written in the form "y = f(x)". Instead, the mapping is from the space ℝ2 to the zero element in ℝ (just the ordinary zero 0): and is an equation in the variables. Implicit functions are a more general way to represent functions, since if: then we can always define: but the converse is not always possible, i.e. not all implicit functions have the form of this equation. One-dimensional space curves in ℝn Formulation Given the functions