Let $g(x)$ be a continuous real valued function defined on $[-a,1+a]$ where $a>0.$
Let $f(x)$ be a continuous real valued function, $f\left( x\right) \geq 0$ for $x\in \lbrack 0,1],$ equal to zero otherwise.
Let $g_{n}(x)$ be a continuous real valued function with domain $% [-a,1+a].$
Let $g_{n}\left( x\right) \rightarrow g\left( x\right) $ for any $x\in \lbrack -a,1+a],$ so that $g_{n}$ converges to $g$ pointwise.