Unbounded values approaching a point prevent continuous extension

Statement

Let $u$ be a function with values in a finite-dimensional normed space. If points $(x_j,t_j)$ in its domain satisfy

(x_j,t_j)\to(x_*,T),\qquad |u(x_j,t_j)|\to\infty,

then $u$ has no continuous extension to a neighborhood of $(x_*,T)$ agreeing with its original values.

Proof and comparison use

Continuity of an extension would give $u(x_j,t_j)\to u(x_*,T)$ , a finite value. Equivalently, a continuous extension is bounded on a sufficiently small compact neighborhood. Thus a divergent sequence inside that neighborhood is impossible.

If a uniqueness theorem forces a proposed extension or competing solution to agree with $u$ before $T$ , the same sequence rules out that competitor. The uniqueness statement and its hypotheses must be supplied separately. The spatial points must remain near a finite point; growth only along points escaping to infinity gives no such local contradiction.