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Can we construct a sequence of functions in the space $C_b(\mathbb{R})$ (continuous and bounded functions) such that

$$ f_n(x) \to f(x) \ \ \forall x \in \mathbb{R} $$

where $f \in C_b(\mathbb{R})$ with the additional property that $$ f_n(1/n) \not \to f(0) $$

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Yes, there is. Take $f_n$ to be a function whose graph is a triangle based at the vertices $(0,0),(0,2/n)$ and $(1/n,n)$, and zero elsewhere. Clearly $f_n$ is continuous for each $n$, and for every $x\in\mathbb{R}$ the sequence $f_n(x)$ tends to zero, but $f_n(1/n)=n$.

If you want a sequence of uniformly bounded continuous functions, change the vertex $(1/n,n)$ to $(1/n,1)$, so that $f_n(1/n)=1$ for all $n$.

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