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Today's calculation of Integral 156

Source: Rikkyo University entrance exam/Science 1990

October 2, 2006
calculusintegrationfunctionlimitderivativealgebrafunctional equation

Problem Statement

For arbiterary integers n,n, find the continuous function f(x)f(x) which satisfies the following equation. lim⁡h→01h∫x−nhx+nhf(t)dt=2f(nx).\lim_{h\rightarrow 0}\frac{1}{h}\int_{x-nh}^{x+nh}f(t) dt=2f(nx). Note that xx can range all real numbers and f(1)=1.f(1)=1.
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