Is there an example of a function $f:(a,b)\times(c,d)\to\mathbb{R}$, which is real analytic in its domain, integrable in the second variable, and such that the function 
$$ g:(a,b)\to\mathbb{R},\qquad g(x) = \int\_c^d f(x,y) dy$$
is not real-analytic on $(a,b)$?


**Edit:** What about an example of bounded $f$ satisfying the above?



Can integration spoil real-analyticity?
$$\int\_0^1 \sqrt{x^2+y}\; dy = \dfrac{2}{3} \left((x^2+1)^{3/2} - |x|^3\right)$$
for $x \in (-1,1)$.


