6 Lamb, on Continuity. 



without exception, belong to one or other of two mutually 

 exclusive categories ; either they have to the left of them 

 points for which <j>(x) is negative, or they have not. More- 

 over, every point of the former category lies to the right 

 of every point of the latter. Hence there must be some 

 point C between A and B, such that every point to the 

 right of C belongs to the former category, and every point 

 to the left of it to the latter. Further, at the point C itself, 

 the value of <p{x) must be zero. For if it were positive, 

 then, in virtue of the continuity, there would be points to 

 the right of C for which <p(x) is positive ; and if it were 

 negative there would be points to the left of C for which 

 <p(x) is negative. Either of these suppositions is incon- 

 sistent with the above determination of C. 



It follows, in the usual manner, that a continuous 

 function cannot change from one value to another without 

 passing once (at least) through every intermediate value. 



II. In every finite range of the independent variable, a 

 continuous function has a greatest and a least value. 



More precisely, if </>(x) be a function which is con- 

 tinuous from x = a to x=b, inclusive, and if /x be the 

 upper limit of the values which <p(x) assumes in this 

 interval, there is some value of x in the interval for which 

 <j)(x)=iul. Similarly for the lower limit. 



To those who are accustomed to associate with every 

 function a graphical representation, the theorem may 

 appear to be self-evident. If the matter be reviewed after 

 studying a rigorous proof, or (better still) after attempting 

 to construct an independent proof, it will be seen that the 

 fallacy (for such it is) in this supposition is due to the 

 fact that every line which we can draw on paper, or even 

 mentally picture to ourselves, has a certain breadth.* 



* See F. Klein, Ueber den allgemeinen Functionsbegviff, itnd dessen Darstelliing 

 durch eine willkurliche Curve, Math. Ann , vol. 22, p. 249. 



