6 i/.JT/C GROAtKTRY [On. 1. 



ways are usually denoted by a single letter having accents 



fcttl ::.': kbm /. f i ''". ' d*note variai. les thiit are 

 similarly involved in a given problem. 



Again, each of the two equations, ^ = 83* 4z+10 and 

 y(x), asserts that y is a function of x\ hut while tin- 

 i.lls precisely how y depends upon x, the latin 

 merely asserts that there it such a dependence, \\ithom 

 giving an v information concerning the form of that <lejenl- 

 ence. If several different forms of functions present them- 

 selves in the same prohlcm. they are represented by different 

 letters, each letter representing a particular form for that 

 prohlein, though it may be chosen to represent an entirely 

 different form in another prohlein. 



;., if the form of <, in a given problem, is defined by the equation 



,>-'-i=^ 



then, in the tame problem, 



= ' and 



7. Continuous and discontinuous functions. In general a 

 function takes different values when different values are 

 assigned to its independent variable. If y=<f>(x), then, 

 r = a and x = b* the function becomes y l = <(a) and 

 y f = (&), and y l is in general different from y v The func- 

 tion <fr(x) is said to be a continuous function of x between 

 x = a and x = b, if, while x is made to pass successively 

 through all real values from a to 6, y remains real and finite 

 and passes correspondingly through all values from y 1 to y 2 . 



This definition may be more precisely stated, thus : If x, and x 2 are 

 any real valve* of x which lip U-tween the value* n :ui<l //. and if the cor- 

 of y, viz. ^(Xj) and <f>(x t ), are real and finite ; and if 



