8 !/>//< GEOMETRY [Cn. I. 



small, i.r. t la taking ;r, sufficiently near x r y a can !>< made 

 to differ from */, by less than any assigned numl>er (t ). how- 

 ever small. Hut this is ; for 9 may be taken as near 



zero as desired, hence the facto r ' ./ , *- -I + ;> ?/ as |1( ' :ir Oxj + 4 

 as desired, and the product therefore as near zero as is neces- 

 sary to be less than e. 



On tin- otli-T ii.tml. if, at regular intervals of time, apples 

 are dropped into a basket, the combined weight of tin* !>,. 



apples will increase discontinuously ; i.e., their total 

 weight is a discontinuous function of the time. 



EXERCISES 



1 If Ax + By + C = 0, prove that y is a continuous function of ar; 

 and x, of y. 



2. If x*+ y*- 4=0, prove that y is a continuous function of x, when 

 2>x>-2. 



3. If ^i + *5= 1, prove that x is a continuous function of y, when 



a 1 or 



4. If -i-^-l=0, isza continuous function of y? 



Or tr 



5. If *f - 9 = 0, is * a continuous function of / ? 



6. If u - 3 v = 0, is u a continuous function of i ? Is v a continu- 

 oufl function of 



7. Show that all functions of the form 



af -I- a,*"- 1 -- ajr"-* + - + a,,_ix 4- a*, 



w)ire a^ a,. ,-, are constants, are continuous for all finite values 

 of x. 



I i 



8 If y _ = V ' ', show that y is discontinuous for x - 1. 



i 

 9. Find the value of x for which y, = c ~ , is discontinuous. 



