$f:[0,1]\rightarrow\mathbb{R}\in C[0,1]$ show $\lim_{a\rightarrow 0^+}(\int^1_a(t^{\frac{-1}{2}}f(t))\,dt)$ exists $f:[0,1]\rightarrow\mathbb{R}\in C[0,1]$ (is continuous) 
Show that
$$\lim_{a\rightarrow 0^+}(\int^1_a(t^{\frac{-1}{2}}f(t))\, dt)$$ exists
I've thought about using integration by parts, but the problem of 1/sqrt(0) still happens and I don't think continuity is enough to change that.

Since $f$ is continuous on $[0,1]$ so are the functions $g: [0,1]\to \mathbb{R},\ t \mapsto f(t^2)$ and $G:[0,1] \to \mathbb{R},\ G(x)=\int_x^1g(t)\,dt$.
Setting
$$
u=\sqrt{t},
$$
we have
$$
\int_a^1t^{-1/2}f(t)\,dt=2\int_{\sqrt{a}}^1f(u^2)\,du=2\int_{\sqrt{a}}^1g(u)\,du=2G(\sqrt{a}).
$$
Hence
$$
\lim_{a\to0+}\int_a^1t^{-1/2}f(t)\,dt=2\lim_{a\to0+}G(\sqrt{a})=2G(0).
$$
