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Two different function sequences $\{f_n\}$ and $\{g_n\}$ both converge to a function $h : (a,2a) \to \mathbb{R}$ uniformly in $(a,c-\delta) \cup (c+\delta,2a)$ for any $0 < \delta < c-a$ and convergence pointwise to $h(c)$ at $x = c$. I'd like to know if it is implied that $\{f_n - g_n\}$ converges to zero uniformly in (a,2a) ?

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up vote 3 down vote accepted

No. You can take the counterexample given in your previous question.

Take $f_n(x) = x^n$ on $[0,1]$ and $f_n(x) = (2-x)^n$ on $[1,2]$ and take $g_n(x)=h(x) = 1$ if $x=1$ and $0$ everywhere else.

Oh, and then compose all your functions with some affine function that maps $[0,2]$ to $[a,2a]$ and forget about the points $a$ and $2a$ to get the domain you want.

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