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[1FZ] Prerequisites:Riemann integral,[19Q].Let \(IβŠ† ℝ\) open interval with \(0∈ I\). Given \(f=f(x,y):IΓ—[0,1]→ℝ\) continuous, and such that also \(\frac{βˆ‚ }{βˆ‚ x} f\) exists and is continuous, set

\[ g(x)=∫_ 0^ 1 f(x,y) \, {\mathbb {d}}y\quad , \]

show that \(g\) is of class \(C^ 1\), and that

\[ g'(x)=∫_ 0^ 1 \frac{\partial ~ }{\partial {x}} f(x,y) \, {\mathbb {d}}y~ . \]

Solution 1

[1G0]

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Authors: "Mennucci , Andrea C. G." .
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