I was reviewing my text from Real Analysis, and something occured to me that hadn't before, nor is it mentioned in the text. The way you usually show that a sequence $\{f_n\}$ of functions does not converges uniformly is to first find a candidate function $f$ to which $\{f_n\}$ converges pointwise. Then, since uniform convergence implies pointwise convergence, if the the function does not converge uniformly to $f$, it cannot converge uniformly to any $f$. My question is regarding uniqueness of the limits.
Uniform convergence is equivalent to convergence in $(X,\lVert\cdot\rVert_{\infty})$, so there is no question of uniqueness of limits in metric space.
But I'm not sure about pointwise convergence. Could it be, that other than $f$ there was another test function $g$ that $f_n$ converged to pointwise, but neglected to check that $f_n\to g$ uniformly?