STEP-FUNCTIONS 



7/4 



the fact that the bee at the end of the hour is a little older than it 



was at the beginning : this change is ignored as being too slow. 



Such changes are eliminated by being treated as if they had 



their limiting values. If a single rapid change occurs, it is 



B 



Time — *• 



Figure 7/3/1 : The same change viewed : (A) over one interval 

 of time, (B) overman interval twenty times as long. 



treated as instantaneous. If a rapid oscillation occurs, the 

 variable is given its average value. If the change is very slow, 

 the variable is assumed to be constant. In this way the concept 

 of ' step-function ' may legitimately be applied to real changes 

 which are known to be not quite of this form. 



7/4. Behaviour of step-function form is likely to be seen when- 

 ever we observe a ' machine ' whose component parts are fast- 

 acting. Thus, if we casually alter the settings of an unknown 

 electronic machine we are not unlikely to observe, from time to 

 time, sudden changes of step-function form, the suddenness being 

 due to the speed with which the machine changes. 



A reason can be given most simply by reference to Figure 4/3/1 . 

 Suppose that the curvature of the surface is controlled by a para- 

 meter which makes A rise and B fall. If the ball is resting at A, 

 the parameter's first change will make no difference to the ball's 

 lateral position, for it will continue to rest at A (though with 

 lessened reaction if displaced.). As the parameter is changed 

 further, the ball will continue to remain at A until A and B are 

 level. Still the ball will make no movement. But if the para- 

 meter goes on changing and A rises above B, and if gravitation is 



