985 



with y {(q. An example in which a {«) is everywhere in tiie closed 

 interval (0, A) equal to zero, is provided by dm (.^') = m—'" tr"'% where 

 A = l. 



2"^^. The function a («) is discontinuous at the end-point on the 

 left of (0, .4), and continuous at the end-point on the right of(0, ^). 



Examples: «,„ (a;) = ^ -|- .7;"''. We have R ^ cc, A^l. For «:=0, 

 a {fx) = and in the other points of the closed interval (0,1), a {a) 

 is equal to 1. More generally we may take 



a,n (.r) = o'« + ?/'" {.V 4- .t''«') 

 where c is a positive constant and y=zf{.v) a function of x of the 

 same kind as above, with the restriction that it is for .f = greater 

 than c. The function a («) is then in u = equal to c, and in the 

 other points of the closed interval (0,1) it is equal to ƒ («). so that, 

 if f {0) = c -\- j). where j) is a positive number, it has a saltus at 

 « = on the right, the amount of which is />. 



3'"^. The function a (a) is continuous at the end-point on the left 

 of the interval (0, A), discontinuous at the other end-point. 



a. a («) is Jinite at A; h. a («) is infinite at A. 



Examples of the case 3(7. Take 



dm C'l') = y'" + 'V"^\ 



where y=zf{x) is a function of the kind considered satisfying the 

 further condition that /(1)<:^1; then a(l)=:l and in the other 

 points of the closed interval (0,1) we have a {a) = f{a), so that this 

 function has a saltus at « = 1 on the left of the amount ^, where 



^ = 1 -ƒ(!). 

 If we take y = 0, we have the case that ƒ(«) is in all points 

 of the closed interval (0,1) equal to zero, except at «^1, where 



ƒ(-) = !• 



Examples of the sub-case 3b. Take 



a,n {^) = y'" + m ! A•"'^ 

 where // is a function as regarded without any further restriction 

 being necessary. If we take y :^ 0, a{tx) is zero in the whole interval 

 (0,1) except at « = 1, where n (cc) is infinite. 



4"^''. The function a (a) is discontinuous at both end-points of the 

 interval {0,A): 



a. a {n) is finite at a ^ A. 



b. a [a) is infinite at « =^ A. 

 Examples of 4a. Take 



«,„ {X) = C" + .17/'" + A""2, 



where c is a positive constant less than 1, y z= f^a-) a function of 

 the kind already considered satisfying the further conditions ƒ (0) ^ c 



