I am trying to show an embedding result for a weighted Sobolev space and have come to the following problem:
I have a function $f: (0,a] \rightarrow \mathbb{R} $ such that:
$f$ is bounded and differentiable (classically)
$f'$ is continuous and weakly differentiable
$xf'$ is bounded
$x f''$ is in $L^{2}$ where the second derivative is taken in the weak sense.
Furthermore, $f$ can be approximated by a sequence of functions in $C^\infty [0,a]$ in the sense of the norm:
$\Vert u \Vert^2 = \int_0^a u^2 + \int_0^a (xu')^2 + \int_0^a (x u'')^2 $
I wish to know whether $\lim_{x \rightarrow 0^+} f(x)$ exists. Any ideas?
I know that in order for for the limit not to exist the function must oscillate infinitely often as $x$ approaches 0 and thus the derivative must be unbounded. I cannot however find an example that satisfies all the conditions and cannot seem to get the assumption of the existence of such a function to yield a contradiction.