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Let $f,g :[a,b]\to\mathbb{R}$ be continuous functions such that $$\int\limits_c^df(x)\leq \int\limits_c^dg(x)dx$$ whenever a$\leq$c$<$d$\leq$b.

I need to show that $f(x)\leq g(x)$. I have the idea of using proof by contradiction supposing that $f(x)>g(x)$, but I do not know how to continue.

hardmath
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user189013
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    Then, by continuity, there is a $\delta>0$ so that $f(x)>g(x)$ for all $x$ in the interval $(x-\delta,x+\delta)\cap [a,b]$. – David Mitra Nov 01 '14 at 13:33
  • You can also do this directly: to prove continuity at a point k, take c=k-epsilon and d=k+epsilon as epsilon approaches zero and then apply continuity. –  Nov 01 '14 at 14:13

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maybe this will help, observe that your hypothesis is equivalent to $0\leq\int^d_c g(x)-f(x)dx$, then show that a continuous function such that $0\leq\int^d_c h(x)dx$ for all $c<d$ must be non-negative (by contradiction).

matiasdata
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