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I am trying to show the following result. If $f(x)$ is Riemann integrable when $a_{1}\leq x\leq b_{1}$ and if, when $a_{1} \leq a < b < b_{1}$, so $$\int _{a}^{b}f\left( x\right) dx=\phi \left( a,b\right) $$ and if $f(b+0)$ exists, then $$\lim _{\delta \rightarrow +0} \dfrac {\phi\left( a,b+\delta \right) -\phi \left( a,b\right) } {\delta }=f\left(b+0\right)$$ then, if $f(x)$ is continuous at $a$ and $b$, deduce that $\dfrac {d} {da}\int _{a}^{b}f\left( x\right) dx=-f\left( a\right) $ and $\dfrac {d} {db}\int _{a}^{b}f\left( x\right) dx=f\left( b\right) $.

Thoughts towards the solution Since $f(x)$ is continuous at $a$ and $b$ we know that the following limits exist $$\lim _{\delta \rightarrow -0} \dfrac {\phi\left( a+\delta,b \right) -\phi \left( a,b\right) } {\delta }=-f\left(a -0\right)$$ $$\lim _{\delta \rightarrow +0} \dfrac {\phi\left( a,b+\delta \right) -\phi \left( a,b\right) } {\delta }=f\left(b+0\right)$$

Also i think $\dfrac {d} {da}\int _{a}^{b}f\left( x\right) dx=\int _{a}^{b}\dfrac {df} {da}dx $ and $\dfrac {d} {db}\int _{a}^{b}f\left( x\right) dx=\int _{a}^{b}\dfrac {df} {db}dx $ although i am not sure how to exploit these two ideas to show the result. Any help would be much appreciated.

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