Please let me know in the comments if this question doesn't meet community guidelines so I can reformulate it. Duplicates are welcome.
I've been working on this problem for quite some time - it seems to me like the 'weakest possible hypothesis' to verify that a function is constant. I concluded that the liminf condition allows me to take a sequence $t_n$ tending to zero where the limit evaluated through this sequence is zero, and then use the continuity of $f$ to show that the partial derivatives must exist and be zero. However, I'm struggling to complete this step.
Could someone help me with a hint, solution, or tell me if this approach is wrong? Alternatively, could you suggest another path? I'd appreciate any suggestions or comments that don't require advanced multivariable calculus tools.
Let $\Omega \subset \mathbb{R}^m$ be an open connected set and $f \colon \Omega \rightarrow \mathbb{R}^n$ a continuous function. Suppose there exists a basis $\{u_1, \ldots, u_m\} \subset \mathbb{R}^m$ such that for all $x \in \Omega$,
liminf $_{t \to 0^+} \frac{\|f(x + tu_k) - f(x)\|}{t} = 0 \quad \text{for each } k = 1, \ldots, m.$
Then $f$ is constant on $\Omega$.