Let $f:(0,\infty)\rightarrow \mathbb R$ be a uniformly continuous function on $(0,\infty)$ with the property that for every $c \in (0,\infty)$, $\lim f(nc)=0$ when $n\rightarrow \infty$ ; $n \in \mathbb N$. Then $f$ should be identically $0$. i have a doubt that the question is correct or wrong. i was going through the problem by applying the formal definitions of limits and uniform continuity .but i am totally lost with the problem. please help me anyone
Every continuous function on a compact set is uniformly continuous.
Now, if we assume we are on a compact interval inside $(0,\infty)$, and there is a continuous function which is zero at the endpoints of the interval, then we can extend this function by zero, to a uniformly continuous function on $(0,\infty)$ which of course will not be zero everywhere, but will satisfy the limit criteria, since for any $c \in (0,\infty)$, the sequence $nc$ will eventually be outside of the compact set(compact sets are bounded), and hence $f(nc) = 0$ for all large $n$, so the limit is zero.
For example, take $f(x) = \sin x$ for $x \in [2\pi , 4 \pi]$, and zero everywhere else. Then it is uniformly continuous, satisfies the criteria given but still fails to be zero everywhere.