Suppose that $f$ is Riemann integrable on $[c,1]$ for all $c>0$ and that $\lim_{c\to 0} \int_c^1 f(x)dx$ exists. Additionally, assume that $\lim_{c\to 0} |f(c)|<+\infty$.
Does this mean that $f$ is integrable on $[0,1]$? Intuitively this seems like it should be fine.
I am wondering if this is logically sound because I want to apply the Riemann-Lesbesgue lemma,
which states that for any Riemann integrable function $\lim_{N\to\infty} \int_{-\pi}^{\pi} \sin(Nx)f(x)dx=0$,
to a function integrable on $[-\pi,-\delta]\cup [\delta, \pi]$ for every $\delta>0$. I can show that $\lim_{\delta \to 0} \int_{-\delta}^{\delta} f(x)dx=0$, but does this allow me to use Riemann Lesbesgue Lemma?