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Is the following true?

If $$\int_{0}^{x}f(t)\,dt \leq \int_{0}^{x} c \,dt =cx $$ for all $x>0$, $x$ is real number, and $c$ is some fixed constant,

then

$$f(t) \leq c$$ for all $t>0$?

EDIT: I should said that $f(t)$ is positive function on $t>0$, and $f(t_{1}+t_{2})\geq f(t_{1})+f(t_{2})$, for all $t_{1},t_{2}>0$ if this helps!

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    No,it is not true. Actually,The case that f(t)>c in a countable set is available.2012-03-29
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    Just to be clear, that's a Riemann integral right.2012-03-29
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    To say it more detailed, consider the function f(x)=x in [0,1),f(x)=2 at x=1,and c=12012-03-29
  • 0
    @kuku Since f(x) may not be continuous,the differentiation may not be available.2012-03-29

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