How to check continuity of $f(a) = \int_0^1 \frac{\sin(ax + x^2)}{x}\, dx$ on $[0,1]$? How can I see the continuity of $f(a) = \int_0^1 \frac{\sin(ax + x^2)}{x}\, dx$  on $[0,1]$?
I have no idea how to approach.
Any comment would be very appreciated. 

Let $a, b\in \Bbb R$. For fixed $x\in [0,1]$, the mean value theorem gives $$\sin(ax + x^2) - \sin(bx + x^2) = x\cos(cx + x^2)(a - b),$$ where $c$ is a number between $a$ and $b$. Thus $$|\sin(ax + x^2) - \sin(bx + x^2)| \le x|a - b|.$$ Since this holds for every $x\in [0,1]$,  we have $$|f(a) - f(b)| \le \int_0^1 |\sin(ax + x^2) - \sin(bx + x^2)|\frac{dx}{x} \le \int_0^1 |a - b|\, dx = |a - b|.$$ Since $a$ and $b$ were arbitrary, $f$ is continuous. 
