How does uniform continuity and continuity affect the convergence of a function

Question

Say we are working in $L^{\infty}(\mathbb{R})$

In the below cases what can we say about the convergence?

1) $f_{n},f$ are continuous and $\lim_{n\to \infty}f_{n}=f$

2) $f_{n},f$ are uniformly continuous and $\lim_{n\to \infty}f_{n}=f$

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Furthermore how would the above cases effect the use of the D.C.T on

$$\lim_{n\to \infty}\int_{\mathbb{R}}f_{n}$$