How do u know $T$ mapped on all real numbers. It also can be mapped on some real numbers

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Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $\lim _{n\rightarrow \infty} f’’(x)$ exists for every $x \in \mathbb{R}$, where $f’’(x) = f \circ f^{n-1}(x)$ for $n \geq 2$. Define

$$S=\left\{\lim _{n \rightarrow \infty} f’’(x): x \in \mathbb{R}\right\} \text{ and } T=\left\{x \in \mathbb{R}:f(x)=x\right\}$$

Then which of the following is necessarily true?

- $S \subset T$
- $T \subset S$
- $S = T$
- None of the above

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Let

$h(x) = \lim_{n \rightarrow \infty} f^n(x) = f(\lim_{n \rightarrow \infty}f^{n-1}(x))$

We can perform the last step as the function is continuous for every x. Continuing on,

$h(x) = f(h(x))$ $ \forall x \in R$

because $n\rightarrow \infty\implies n-1 \rightarrow \infty$

It means $S = \{f(x) = x : \forall x \in Range(h(x))\}$

The function defined by two sets in same but $S$ is defined over some elements of real numbers whereas $T$ is defined for all real numbers.

$\implies S \subset T$

$A$ is correct answer.

$h(x) = \lim_{n \rightarrow \infty} f^n(x) = f(\lim_{n \rightarrow \infty}f^{n-1}(x))$

We can perform the last step as the function is continuous for every x. Continuing on,

$h(x) = f(h(x))$ $ \forall x \in R$

because $n\rightarrow \infty\implies n-1 \rightarrow \infty$

It means $S = \{f(x) = x : \forall x \in Range(h(x))\}$

The function defined by two sets in same but $S$ is defined over some elements of real numbers whereas $T$ is defined for all real numbers.

$\implies S \subset T$

$A$ is correct answer.

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