If we were relying on the Riemann theory , we should find a fundamental difficulty here ; for Volterra has shown by an example that may exist everywhere and be bounded , and yet not be integrable in Riemann's sense .sx In the Lebesgue theory , a differential coefficient is measurable , and so integrable if it is bounded .sx But , if it is unbounded , it is not necessarily integrable in the Lebesgue sense .sx The problem has received a satisfactory answer , but it requires a more general process , known as totalization , or Denjoy integration , which we have not space to consider here .sx The result is that if is finite everywhere , then ( 3 ) follows from ( 2 ) if the integral is taken in the Denjoy sense .sx 11.2. Differentiation throughout an interval .sx The ordinary functions of analysis are differentiable in general , i.e. for most values of the variable , though there may be special points at which they are not differentiable .sx The exceptional points are usually isolated .sx This seems to have created the impression at one time that a continuous function necessarily has a differential coefficient in general .sx It was , however , shown by Weierstrass that this is quite untrue .sx There is a continuous function which has no differential coefficient anywhere .sx Nevertheless , the idea that an `ordinary function' has a differential coefficient in general is correct , if we attach this vague expression to a different class of functions .sx We shall see that it is true in the sense that a monotonic function has a finite differential coefficient almost everywhere .sx We shall first consider non-differentiable functions , and then proceed to the constructive side of the theory .sx 11.21. Continuous non-differentiable functions .sx There are many simple examples of continuous functions which are not differentiable at particular points ; for example , if , .sx the ratio tends to different limits , 1 and -1 , as by positive or negative values ; and if , , the ratio does not tend to any definite limit .sx We can next , by a method known as the condensation of singularities , construct continuous functions which are not differentiable in a set which is everywhere dense , for example in the set of rational points .sx Let denote the rational numbers between 0 and 1 , and let .sx where has an assigned singularity at , and the coefficients an tend to zero sufficiently rapidly .sx Then will have the assigned singularity at every rational point .sx For example , .sx is continuous , since the series is uniformly convergent ; but it is not differentiable at any rational point ; for .sx and as the first term tends to a limit , the second term tends to according as or , and , if , the third term does not exceed .sx in absolute value .sx Hence does not exist .sx To obtain functions which are everywhere non-differentiable we have to use quite different methods .sx The first example of such a function was given by Weierstrass .sx 11.22. Weierstrass's non-differentiable function .sx This function is defined by the series .sx where , and a is an odd positive integer .sx The series is uniformly convergent in any interval , so that is every-where continuous .sx On the other hand , if , the series obtained by term-by-term differentiation is divergent .sx This in itself does not prove that is not differentiable , but it suggests possibilities in this direction .sx We shall prove that if , the function has no finite differential coefficient for any value of x. .sx We have .sx say .sx Now , so that .sx .sx We next obtain a lower limit for , giving h a particular value .sx We can write , where is an integer , and .sx Let .sx .sx Then .sx Also .sx Since a is odd , it follows that .sx Again .sx Hence .sx All the terms of this series are positive , and hence , taking the first term only , .sx Hence .sx If , the factor in brackets is positive ; and when , and the expression on the right tends to infinity .sx Hence takes arbitrarily large values , so that does not exist or is not finite .sx The graph of the function may be said to consist of an infinity of infinitesimal crinkles , but it is almost impossible to form any definite picture of it which does not obscure its essential feature .sx 11.23. The following example of a continuous non-differentiable function is due to van der Waerden .sx The function is similar to Weierstrass's , but the result is obtained in quite a different way .sx Let denote the distance between x and the nearest number of the form , where m is an integer .sx Then the function is a continuous non-differentiable function .sx Each is continuous ; and , so that the series is uniformly convergent .sx Hence is continuous .sx Let x be any number in the interval ( 0,1 ) , and suppose it expressed as a decimal .sx If the qth figure is 4 or 9 , let ; otherwise let .sx Then if , the nearest number is the same for x and , and x and lie on the same side of it ; while if , the numbers and corresponding to x and differ by .sx These rules may be verified by considering simple examples , such as , .sx It follows that .sx Hence , where p is an integer , and is odd or even with .sx Hence cannot tend to a finite limit as .sx 11.3. The four derivatives of a function .sx Whether the differential coefficient exists or not , the four expressions always have a meaning , being either finite , or positive or negative infinity .sx They are called the upper and lower derivates on the right , and the upper and lower derivates on the left , respectively .sx We shall denote them by respectively , the sign referring to that of h in the above ratio , and its position corresponding to the 'lower' or 'upper' limit .sx If , the function is said to have a right-hand derivative , if , a left-hand derivative .sx The necessary and sufficient condition for the existence of the ordinary differential coefficient is that all the derivates should be equal .sx We denote the left-hand and right-hand derivatives , when they exist , by and .sx Examples .sx ( i ) The function , where the positive value of the square root is always taken , has different left-hand and right-hand derivatives at x = 0 .sx ( ii ) Let , .sx Then at x = 0 , , , .sx ( iii ) Let , where .sx Then at x = 0 , , , .sx ( iv ) If is continuous in , and one of its derivates is non-negative in the interval , then .sx Let , for example .sx Suppose that , and let .sx Then .sx Also for some sufficiently small values of , since .sx Hence for some values of x between a and b. Let be the greatest such value .sx Then , contrary to hypothesis .sx Hence for every positive , and the result follows .sx ( v ) The derivates and incrementary ratios of a continuous function have the same bounds in any interval ; i.e. if any one of the derivates satisfies , then , and conversely .sx Consider , and use the previous example .sx ( vi ) If one of the derivates of is continuous at a certain point , then has a differential coefficient at the point .sx 11.4. Functions of bounded variation .sx We say that is of bounded variation in ( a,b ) if , in this interval , it can be expressed in the form , where and are non-decreasing bounded functions .sx It is easily seen that the sum , difference , or product of two functions of bounded variation is also of bounded variation .sx An alternative definition is obtained by assuming that , if the interval ( a,b ) is divided up by points , then is less than a constant independent of the mode of division .sx The upper bound of these sums is called the total variation .sx It is easily seen that , if the first condition holds , then so does the second .sx For , so that .sx To prove the converse , let p be the sum of those differences which are positive , is the sum of those which are negative .sx Then , if v is the sum , we have , , and so , .sx Hence , if v is bounded for all modes of division , so are p and n. Let V , P , and N be the upper bounds of v , p , and n. Then , .sx Let , , and be the corresponding numbers for the interval ( a,x) .sx They are obviously bounded non-decreasing functions of x ; and , , so that .sx This is the required expression for .sx The functions , , and are called the total variation and the positive and negative variations of in ( a,x) .sx If is continuous and of bounded variation , its variation is continuous .sx We can find a mode of division of the interval ( a,x ) , with a point of division as near x as we please , such that and also .sx Let .sx Then is a sum corresponding to the interval , and so .sx Since is non-decreasing , it follows that as from below .sx Similarly as from above .sx Hence is continuous .sx A continuous function of bounded variation is the difference between two continuous non-decreasing functions .sx For if is continuous , so are and .sx 11.41. The differential coefficient of a function of bounded variation .sx The object of the next three sections is to prove that a function of bounded variation has a finite differential coefficient almost everywhere .sx Our proof depends on the following lemmas , due to Sierpinski .sx They are of the same type as the Heine-Borel theorem , but apply to sets which need not even be measurable .sx LEMMA 1 .sx Suppose that each point x of a set E in ( a,b ) is the left-hand end-point of one or more intervals of a family H. Then there is a finite non-overlapping set S of intervals of H which includes a sub-set of E such that .sx Let be the set of points of E which are associated with some .sx Then E is the outer limiting set of the sets , we have ( 10.29 ) , and we can take n so large that .sx Let be the lower bound of , its upper bound , and let .sx Let .sx Then there is a point of such that .sx Let be an associated interval for which .sx If there are points of to the right of , let be their lower bound .sx Then there is a point of in .sx Let be an associated interval with .sx Continuing the process , we reach in a finite number of steps , since each step takes us at least nearer to it .sx In fact , if there are N steps , then .sx Let S denote the set of intervals so constructed , and T the set of intervals .sx Then and .sx Hence , and the set has the required property .sx LEMMA 2 .sx Suppose in addition that for every x there are arbitrarily small intervals .sx Then we may conclude in addition that .sx The additional condition is necessary ; we might , for example , take E to be a single point x , and associate with it the interval .sx Then Lemma 1 would hold , but not Lemma 2 .sx Let O be an open set containing E , such that .sx Let be the set formed from O by omitting intervals less than , and shortening the rest by at the right-hand end .sx We can take p so large that .sx Let now denote the set of points of E which are associated with some between and .sx Under the conditions of this lemma , E is still the outer limiting set of the , and we can take n so large that .sx Since , it follows that .sx We can now proceed with in the same way as we did previously with , except that now each of our intervals is less than .sx