Consider a Ito Process,not necessarily Brownian Motion, $(X_t)_{t\geq 0}$ starting at 0 and define the associated Local Time at $0$:
\begin{equation} L_{t}=\lim_{\epsilon \rightarrow 0}\frac{1}{2\epsilon}\int_{0}^{t}\mathbb{1}_{[-\epsilon,\epsilon]}(X_{s})ds. \end{equation}
I want to show that $L_{t}$ is not differentiable at $0$. I suspect that \begin{equation} \lim_{t\rightarrow 0}\lim_{\epsilon \rightarrow 0}\frac{1}{t}\int_{0}^{t}\frac{1}{2\epsilon}\mathbb{1}_{[-\epsilon,\epsilon]}(X_{s})ds \end{equation} this limit is not well defined.
\begin{equation} \lim_{\epsilon \rightarrow 0}\lim_{t\rightarrow 0}\frac{1}{t}\int_{0}^{t}\frac{1}{2\epsilon}\mathbb{1}_{[-\epsilon,\epsilon]}(X_{s})ds \end{equation} is clearly not well defined but I cannot quite justify the switching of limits. Is this the right approach in regards to answering this problem?