For x 0,let h(x) = begin{cases} frac{1}{q} | vedclass

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(A) The function $h(x)$ is known as the Thomae's function (or the popcorn function). $1$. For any rational number $x = \frac{p}{q}$ (where $p, q$ are

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$h(x)$ is discontinuous for all $x$ in $(0, \infty)$.

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(A) The function $h(x)$ is known as the Thomae's function (or the popcorn function). $1$. For any rational number $x = \frac{p}{q}$ (where $p, q$ are relatively prime),$h(x) = \frac{1}{q}$. As $x$ approaches a rational number,the values of $h(x)$ for nearby irrational numbers approach $0$. Since $\frac{1}{q} 
eq 0$,the function is discontinuous at every rational point. Thus,statement $C$ is true. $2$. For any irrational number $x$,$h(x) = 0$. For any $\epsilon > 0$,there are only finitely many rational numbers $\frac{p}{q}$ in any interval such that $\frac{1}{q} \geq \epsilon$. Thus,the limit of $h(x)$ as $x$ approaches any irrational number is $0$. Since $h(x) = 0$ at irrational points,the function is continuous at every irrational point. Thus,statement $B$ is true. $3$. Since the function is continuous at irrational points and discontinuous at rational points,it is not discontinuous for all $x$ in $(0, \infty)$. Therefore,statement $A$ is false. $4$. Since the function is discontinuous at every rational point,it is not differentiable (derivable) at any rational point. Thus,statement $D$ is true. Conclusion: Statement $A$ does not hold good.

For $x > 0$,let $h(x) = \begin{cases} \frac{1}{q} & \text{if } x = \frac{p}{q} \text{ (where } p, q \in \mathbb{N} \text{ are relatively prime)} \\ 0 & \text{if } x \text{ is irrational} \end{cases}$. Which of the following statements does not hold true?

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