4 Lamb, on Continuity. 



increment of y. Then if, or being any positive quantity 

 different from zero, we can always find a positive quantity 

 €, different from zero, such that for all admissible values 

 of ox which are less (in absolute value) than e, the value 

 of Sy will be less in absolute value than cr, the function 

 is said to be " continuous " for the particular value x of 

 the independent variable. 



As already indicated, this test can be easily applied to 

 particular cases ; but it does not lend itself very readily to 

 the proof of general theorems. The reason is that the 

 definition does not warrant us in making any statement 

 whatever about the value of the function for any value of 

 x other than the one referred to, however near. In the 

 ignoring of this consideration lies a fruitful source of 

 fallacies. 



4. We may construct a mental representation of the 

 relation between two variables x, y ; one of which is a 

 function of the other, by taking rectangular co-ordinate 

 axes X'O X, Y'O Y. If we measure OM along OX to 

 represent any particular value of the independent variable 

 x, and ON along OY to represent the corresponding 

 value of the function y, and if we complete the rectangle 

 OMPN, the position of the point P will indicate the 

 values of both the associated variables. 



Y 











K 









N 

 H 





o 









] 



O 



A 



IS 



r\ 



B 



Since, by hypothesis, M may occupy any position on 

 X'X, between (it may be) certain fixed termini .4, B, we 



