Problem 1374 (difficulty: 7/10)

Let \(\displaystyle f:\R^2\to\R\) be continuous and \(\displaystyle \displaystyle G(x)=\int_{-x}^{x^2} f(x,y)\dy\).

(a) Prove that \(\displaystyle G\) is continuous.

(b1) Show that if \(\displaystyle f\) is continuously differentiable, then \(\displaystyle G\) is also continuously differentiable. What is \(\displaystyle G'\)?

(b2) Can the condition of continuously differentiability weakened?

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