$\displaystyle \mathop {\lim }\limits_{h \to 0} A(x + h) - A(x) = \mathop {\lim }\limits_{h \to 0} hf(x)\\f(x) = \frac{{\mathop {\lim }\limits_{h \to 0} A(x + h) - A(x)}}{h} = \frac{{dA(x)}}{{dx}}\\$

this only proves that derivative (or tangent ) to area function is function itself....

how to prove that area function is the integral function ..

hereA(x)is the area under the curve f(x)....