Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

THE MATHEMATICS OF PSYCHOLOGY 379
function of stimulus within the interval between the
initial and the terminal thresholds of any department.
For to any amount of stimulus in such an interval, there
corresponds a sensation. That is to say, in no such
interval have there been found any blind spots or gaps
or regions or points where the sensorium fails to respond.
The question arises: Is sensation, or the intensity of sensation, a continuous function of stimulus? Most psychologists answer yes. Among these may be mentioned, for
example, Titchener, Ward and Stout. James has
answered no. The men named leave one in doubt
whether they know precisely what is meant by a continuous function. Let us recall to mind the idea of
functional continuity. Let us remember that if f(x) is
to be a continuous function of a real variable x in an
interval having a for its beginning and b for its end, the
following conditions must be satisfied:
(I) If x' be any value of x in the interval, then f(x') =
some definite value.
(2) Limit f(x'z-Ax)=f(x') as Ax approaches zero.
Condition (I) is indeed included in (2) but it is helpful to
state it explicitly. Now we know that the ordinary function, y=c logx, is continuous throughout any interval
not containing the value, x =0. But is the Fechner function, S=C log R, a continuous function of R? No; for
consider a stimulus greater than A(I+r)" and less than
A(I +r)f+l; compare the corresponding sensation S with
that denoted by n (in the above-given table) and then
compare it with that denoted by n + I; in the first comparison S appears to be n; in the second, it appears to be
n+I. Condition (I) is, you see, not satisfied. Then, of
course, condition (2) is not satisfied. Moreover, when we
speak off(x) as a continuous function, we imply that the