The number of continuous functions f:[0,1] ri | vedclass

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(A) Given that $f(x) < x^2$ for all $x \in (0,1]$.Integrating both sides from $0$ to $1$:$\int_{0}^{1} f(x) dx < \int_{0}^{1} x^2 dx$.We know that $\i

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$0$

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(A) Given that $f(x) Integrating both sides from $0$ to $1$:$\int_{0}^{1} f(x) dx We know that $\int_{0}^{1} x^2 dx = \left[ \frac{x^3}{3} \right]_{0}^{1} = \frac{1}{3}$.Therefore,$\int_{0}^{1} f(x) dx However,the problem states that $\int_{0}^{1} f(x) dx = \frac{1}{3}$.This creates a contradiction because the integral of a function strictly less than $x^2$ must be strictly less than the integral of $x^2$.Thus,no such continuous function $f$ exists.The number of such functions is $0$.

The number of continuous functions $f:[0,1] \rightarrow [0,1]$ such that $f(x) < x^2$ for all $x \in (0,1]$ and $\int_{0}^{1} f(x) dx = \frac{1}{3}$ is:

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