C onver gency of Fourier s Series. 137 



a small straight element inclined at a definite angle to the 

 axis or ,v. 



25. A function which is differentiable wherever it is con- 

 tinuous is said to possess ordinary continuity. We thus see 

 that ordinary continuity is only a particular kind of continuity. 

 It is, however, the kind exclusively dealt with in the Infini- 

 tesimal Calculus ; for the processes of the Differential Calculus 

 are based upon the properties of the differential coefficient, 

 and, practically at least, integration is treated as the inverse 

 of differentiation. While, however, every finite and con- 

 tinuous function has an integral, only some possess a differen- 

 tial coefficient. Here, then, the inverse operation is always 

 admissible (though it cannot always be formally effected), 

 whereas the direct operation is not always admissible. For 

 this reason Weierstrass, in his lectures, once made the definite 

 integral the starting-point for the investigation of the pro- 

 perties of functions, and especially of the condition for the 

 existence of a differential coefficient. 



26. Examples of functions which are continuous and per- 

 fectly determinate, but not differentiable, were first given by 

 Weierstrass*. The essential feature in the case of such func- 

 tions is that the loci consist of an infinite number of infinitely 

 small zigzags and oscillations (for otherwise the functions 

 would be differentiable). The functions are thus perfectly 



determinate and continuous ; but *__ n — — =—^ — 



cannot anywhere have a determinate value, and the processes 

 of the Differential Calculus are therefore inapplicable. When 

 drawn the locus of a function of this kind is indistinguish- 

 able from that of a function having ordinary continuity, and 

 whose values at the different points are the mean of the 

 values of the given oscillating function at the same points. 

 But we cannot treat the two as analytically the same. Thus, 

 to borrow an illustration used by Prof. Greenhill, the zigzag 

 locus C D is indistinguishable from the straight line A B 

 when the zigzags are infinitely small and infinitely nu- 

 merous. But we cannot treat it as having the properties of a 

 straight line. For the length of the zigzag locus is always equal 

 to the sum of the lengths of C E and E D, however small we 

 make the zigzags, provided they do not alter in form. If, then, 

 we treat the zigzag locus as a straight line when the zigzags 

 are infinitely small and infinitely numerous, it follows that 

 the third side of a triangle is equal in length to the sum of 



* Cayley's article " Function," Encyc. Britt. 



