Suppose $f: [-1,1] \rightarrow \mathbb{R}$ be a Riemann integrable function that is continuous at x=0. Assume $\phi_{n}(t)$ is a sequence of nonnegative Riemann integrable functions on [-1,1] satisfying the following two properties:
(a) $\int_{-1}^{1} \phi_{n}(t)=1$ for every n.
(b)For every $\delta$ greater than zero we have that $\phi_{n} \rightarrow 0$ uniformly on $[-1,-\delta] \cup [\delta,1]$ as $n \rightarrow \infty$. Prove that:
$\lim_{n \rightarrow \infty} \int_{-1}^{1} f(t)\phi_{n}(t)dt=f(0)$
Now, I used the unfiorm continuity of $\phi_{n}$ to show that
(c) $\lim_{n \rightarrow \infty} \int_{-\delta}^{\delta} \phi_{n}(t)dt=1$,
since we can swap the limit with intgeral. Then I broke down the integral in the question to the three parts. Since $f\phi$ is not necessarily uniformly continuous, we do not have that the limit of their integral from -1 to $-\delta$ is zero, same holds for $\delta$ to 1. And for the middle integral I wish that I could use the c, but again $f\phi$ is not necessarily uniformly continuous. Maybe I should take a totally different approach. Any help would be appreciated.