Sequence of continuous functions convergent to $0$ with the integral equal to $1$

Question

I am looking for a sequence of continuous functions $\{f_m\}$ defined in $A\subset\mathbb{R}$ with $\lim\limits_{m\to\infty} f_m=0$ such that $\int_A f_m \;d\mu=1$.

The problem I have is with the continuity. If $\{f_m\}$ is not continuous I can find the example.

Answer 1

Let $A = (0,1)$ and $$f_n(x) = (n+1)x^n$$ Then $f_n(x) \to 0$ pointwise and $\displaystyle \int_A f_n \ d\mu = 1$.