THE TEN PILLARS 11 



These functions are all continuous; that is, at no point does the slope change 

 suddenly from one value to another. It is probable that there are no discon- 

 tinuous functions in nature, although the change in slope may be so sharp as 

 to seem discontinuous in the first and cursory observation. Thus, phe- 

 nomena involving the interface or juncture of two phases, as for example at 

 the cell wall, are examples of rapidly changing continuous functions which 

 at first sight appear to be discontinuous. 



3. Limits 



If a variable, changing in accordance with some assigned law, can be 

 made to approach a fixed constant value as nearly as we wish without ever 

 actually becoming equal to it, the constant is called the limiting value or limit 

 of the variable under these circumstances. 



A circus abounds with examples in which exceeding a limit in either dis- 

 tance or time would mean a severe penalty. Consider the "hell drivers" who 

 ride motorcycles inside a 40-ft cylinder, approaching the top — the limiting 

 height — as closely as they dare, yet never suffering the disaster of actually 

 reaching it. In other words, if y = /(*), and if, as x approaches a, y ap- 

 proaches some value, b, then b is said to be the limit of/(x) when x equals a. 

 In shorthand, for the functional relationship y = f(x), if x — * a as y — * b, 

 then 



Lim f(x) = b 



x — '0 



It is often useful to approach a limiting value and study its properties 

 without having to suffer the embarassment sometimes associated with the 

 limit itself. This concept was introduced by Leibnitz 300 years ago. 



4. Increments 



A small fraction of any quantity under observation is called an increment. 

 Increment is thus exactly translated as "a little bit of." It is given a symbol, 

 the Greek letter delta, A. 



As the variable, x, increases (Figure 1-3) from zero to high values, that 

 amount of x between A and B (i.e., x 2 — x x ) is "a little bit of" x, and is 

 written in shorthand: \x. 



!— *f 



i 



i 



i r 



A P B 40mph 



Figure 1-3. Increments of Distance and Time, 

 Ax and Af, used in defining velocity, Ax/Af, 

 abouf point P, or dx/dr ai point P. 



