Updated S2013 Functions and Continuity Background FUNCTION. ======================= A function is a rule or equation that defines one value for each instance of its variables. For example, y=1+x defines one y for each value of the variable x. And z=x+y+1 defines one value z for each instance of variable values x,y. FUNCTION NOTATION. ======================= Function notation f(x) or f(x,y) or f(x,y,z) was introduced by Peter Gustav Lejeune Dirichlet (1805-1859). Historical remarks about Dirichlet and be found at Wikipedia: http://en.wikipedia.org/wiki/Johann_Peter_Gustav_Lejeune_Dirichlet The variable names x,y,z are supposed to be dummy variables subject to substitution of actual values. For example, if the defining equation is f(x,y) = x + y, then f(0.1,0.7) means Substitute x=0.1 and y=0.7 into the equation x+y, simplify, and report the numerical answer. We would write f(0.1,0.7)= (0.1) + (0.7) = 0.8 and say f of 0.1, 0.7 equals 0.8 This same fundamental idea applies to modern computer languages like C++. Here, a function is a subroutine which computes some answer, to be used elsewhere. An example: float f(x,y) float x,y; { return (x+y); } The variable names x,y are dummy variables representing actual floating point numbers. For instance, f(0.1,0.7) represents 0.8, the computer calculation exactly parallel to the mathematical discussion above. CONTINUITY. ======================= A working definition for most applied work is that a function is continuous provided we can legally find its derivative(s). Then all the elementary functions of calculus are continuous, because they appear in the derivative table. An example of a known continuous function that cannot be justified to be continuous based upon differentiation is |x|. We can't legally take its derivative at x=0. However, continuity of |x| can be justified by other means, for example, by viewing it as the pasting together of the differentiable functions x and -x. There are useful theorems about continuity that apply to decide if some complex expression is the equation of a continuous function. These theorems also decide for |x+1|, an expression that is subtly simple. Some notable theorems: The sum of continuous functions is continuous. The same is true for products and quotients. Compositions of continuous functions are continuous. To elaborate on these statement, an expression z = (x+2y)^3/(1+x^2y^2) is continuous, because it is the quotient of continuous functions (x+2y)^3 and 1+x^2y^2. The power (x+2y)^3 is the composition of u^3 and u=x+2y, both of which are continuous (because they are differentiable). The polynomial 1+x^2y^2 is differentiable (in variables x,y) and therefore continuous. Functions like z=1/(x+y) have a natural domain of definition, namely where the fraction is defined, in this case where x+y is nonzero. Because we can legally find its partial derivatives on this domain, we justify it is continuous there. A function like z=(y-1)^(1/3) makes sense everywhere and because it is a composition of u^(1/3) and u=y-1, then it is justified to be continuous everywhere by calculus theorems (see above). It cannot be justified to be continuous everywhere by taking partial derivatives, because the y-partial of z has y-1 in the denominator, causing it to be infinity at y=1. However, the partial derivative argument applies at all other x,y (y not equal to 1). A similar example is z=y^(1/2) in the box defined by 0