Define a sequence of functions as follows: $$f_k(x) = \begin{cases} 2k& x \in[\frac{-1}{k},\frac{1}{k}] \\ 0&x \in[-1,1]\setminus[\frac{-1}{k},\frac{1}{k}] \end{cases}$$
Does ${f_k}$ converge in $C[-1,1]$ to a continous function? What can be said about the integrals for g(x) being polynomials $$\int_{-1}^{1}g(x)f_k(x)dx$$ as $k$ approaches infinity?
Attempt:
The function clearly approaches something that looks like a dirac delta function. Since this is unbounded as $k$ approaches infinity, the function does not converge in $C[-1,1]$. My problem is with what can be said about the integrals. All polynomials are bounded on $[-1,1]$ which suggests that all the integrals are bounded as well. At infinity, it seems like the value of the integral will be $4g(0)$. But both these reasoning seems a bit thin. What else can be said about the integrals as $k$ approaches infinity?