Let $f: [0,1] \to \Bbb R^+$ be continuous map then is it possible to have $\int_0^x f(t)dt \geq f(x)$? Let $f: [0,1] \to \Bbb R^+$ be continuous map then is it possible to have $\int_0^x f(t)dt \geq f(x)$? If such functions exists then what will be the cardinality of the set having these kind of functions?
If I have $f$ is differentiable then we get $f(x) \leq e^x$. But for $e^x$ the condition will not be satisfied. So how can we proceed? 
 A: The cardinality of the set of such functions is 1 :-)  The only such function is $f \equiv 0$.
Let $u(x) = \int_0^x f(t)\,dt$.  Then $u'(x) = f(x)$ so we have the differential inequality $u'(x) \le u(x)$.  By Gronwall's inequality with $\beta(x) =1$, we conclude $u(x) \le u(0) e^t = 0$.  Since $f \ge 0$ we also have $u \ge 0$.  Hence $u \equiv 0$.  Differentiating, $f \equiv 0$ as well.
Actually, the proof of Gronwall's inequality reduces to a very simple argument in this case.  Let $g(x) = u(x) e^{-x}$.  Then the product rule gives $g'(x) = (u'(x) - u(x)) e^{-x} \le 0$.  So $g$ is decreasing.  $g(0) = 0$ so we have $g \le 0$.  Since $e^{-x} > 0$, we have $u \le 0$ and proceed as above.
A: Since $[0,1]$ is closed and $f$ is continuous, $m = \min\limits_{x \in [0,1]}f(x)$ and $M = \max\limits_{x \in [0,1]}f(x)$ exist. 
Since $f : [0,1] \to \mathbb{R}^+$, we must have $0 < m \le M$. 
Now, can you find a value of $x \in [0,1]$ for which $\displaystyle\int_{0}^{x}f(t)\,dt < f(x)$?
A: The property
$$
\int_0^x f(t)\,dt \ge f(x),\quad x\in [0,1],\tag{1}
$$
is equivalent to
$$
\mathrm{e}^{-x}\left(\int_0^x f(t)\,dt - f(x)\right)\ge 0,\quad x\in [0,1],
$$
or
$$
\left(\mathrm{e}^{-x}\int_0^x f(t)\,dt\right)'\le 0,\quad x\in [0,1].
$$
So if $g: [0,1]\to\mathbb R$, is continuously differentiable and decreasing, and $g(0)=0$, then $\,f(x)=\big(\mathrm{e}^{x}g(x)\big)'\,$ satisfies $(1)$. 
Example. $f(x)=-(x+1)\,\mathrm{e}^{x}$. 
General form of $f$. All these functions are of the form
$$
f(x)=\mathrm{e}^x\big(g(x)+g'(x)\big)=-\mathrm{e}^x\left(h(x)+\int_0^x h(t)\,dt\right),
$$
where $h$ is continuous and non-negative.
