= Solution
\b[No.] Consider the triangular functions $f_n(x)=\max(1-n|x|,0)$. The absolute-value and positive-part maps are <Lipschitz continuous>, so $|f_n(x)-f_n(y)|\leq n|x-y|$. Each $f_n$ is therefore <uniformly continuous> on the whole real line. For $x=0$ its value is always one; for every fixed $x\ne0$ it eventually vanishes. Hence its <pointwise limit> is
$$
f(x)=\begin{cases}1,&x=0,\\0,&x\ne0.\end{cases}
$$
This limit is not <continuous> at zero: along $x_j=1/j\to0$, $f(x_j)=0\not\to1=f(0)$. Individual <uniform continuity> does not provide a modulus uniform in $n$.
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