.11, 12] VARIABLES AND FUNCTIONS. 19 



m 



e.g. the function u = - has 



y 



lim j lim u I = lim (0) = 0, 



y = ( 3 = ) 2/=0 



since the limit for x does not contain y, while 

 lim j lim u \ = oo . 



#=0 ( 2/=0 ) 



A function of two or more variables is continuous at a point 

 x = a, y = b, if for any positive value of e, however small, we can 

 find Sj and 8 2 so that 



h, y + k)-f(x, y) | <e, | h | <8 lf | & | < S 2 

 for aW values of h and & which satisfy the above inequalities. 



A function of two variables is not necessarily continuous if it 

 is a continuous function of either variable; e.g. the function 

 gey I (a? + y 2 ) is a continuous function of x for any value of y, even 

 y= 0, and of y for any value of x, even x = 0. It is not a continuous 

 function of x and y at x = 0, y = 0, since u = for a? = 0, irrespective 

 of the value of y, and u for y = 0, irrespective of the value of 

 x, but if we select pairs of values of x and y, such that y = m#, we 



77? 



have u = ^ -- 5 , which is discontinuous with the value ^ = at 

 1 +m 2 



Derivative. If u considered as a function of x, for any par- 

 ticular value of y, has a derivative as before defined, this derivative 

 is called the partial derivative of u with respect to x, and is 

 denoted by /a/ or by 



, . 



# fc=0 



)/* 

 If %- considered as a function of y, say < (y), has a derivative, 



Cw 



r)f 



this is called the partial derivative with regard to y of - , and is 



ju 9 V 



denoted by ^- = lim 



* 



22 



