d-7.] WTRODUn T 



a positive number oao be found, iuch thai by taking numencaDy, 



:l follow that, numerically* 



where c u any aetigned positive number, however small; then *Xr) ie a 

 . noun fuiu'tioi. values from a too. 



in worth: y U a continuous function of x for all value* of x in the 

 taking any two value* of x in th interval 



ufflcif ntly near together, the difference between the corresponding values 

 of y can be made leai than any attigned number, however miiaJI. 



A discontinuous function is one that does not fulfil the 



us for continuity. It in, however, Dually discon- 



tinuous for only ;i limited numher of particular values of its 



I undent variable, while between these values it is con- 



tinu< 



As f.uniliair examples of continuous functions may be 

 mentioned : the length of a solar shadow ; the area of a 

 cross-section ,,f a growing tree, or of a growing peach : 

 height of the mercury in a barometer ; the temperature of a 

 room at varying distances from the source of heat ; and 

 interest as a function of time. 



So, also, y = 8x* + 4x+l is a continuous function of x 

 for all finite values of jr. 



I or, y remains real and finite so long as x remains real and 

 finite, ;inL if r, and z, be any two finite values of x which 

 differ from each other by 17, t.<., if r 2 = T } 17, then 



- y i - 



-SCr, ^) + 4 O, ^) + 1 - Olr,' 4. 



Now to show that y3i j + 4x + l is continuous for 

 9 = JT V it only remains to show that, by taking r; sutVu iently 



