The function is continuous on (0,1/2]. But it

has neither a maximum nor a minimum on that interval . From

these graphs, you can see that if we consider the portion of

the graph on an interval [c,1/2] with c > 0, then

* the closer c is to zero, the bigger the maximum and

the smaller (more negative) the minimum of f(x) are, BUT

* there are still infinitely many oscillations of the function

in the interval (0,c), so the growth never stops.

N.B. Look carefully at the vertical scales on these graphs

first: c = .1

> plot(sin(1/x)/x,x=.1..0.5);

next, c = .05

> plot(sin(1/x)/x,x=.05..0.5);

Finally, c = .01:

> plot(sin(1/x)/x,x=.01..0.5);

>