Let
. Note that since
is differentiable (and since
is differentiable), then
is also differentiable (and therefore continuous).
Let
, then
.
Let
, then
.
Thus, by the intermediate value theorem (which applies because, as we saw,
is continuous), there exists
such that
.
Using the same method, by the periodicity of
we see that for any integer
, there exists
such that
.
.