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?

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