Let $f$ and $g$ be two functions from $[0, 1]$ to $[0, 1]$ with $f$ strictly increasing. Which of the following statements is always correct? If $g$ is continuous, then $f ∘ g$ is continuous. If $f$ is continuous, then $f ∘ g$ is continuous. If $f$ and $f ∘ g$ are continuous, then $g$ is continuous. If $g$ and $f ∘ g$ are continuous, then $f$ is continuous.

Let $X \subset \mathbb{R}$ and let $f,g : X \rightarrow X$ be a continuous functions such that $f(X)\cap g(X)=\emptyset$ and $f(X)\cup g(X)= X$. Which one of the following sets cannot be equal to $X$? $[0, 1]$ $(0, 1)$ $[0, 1)$ $\mathbb{R}$

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a continuous function and $A \subset \mathbb{R}$ be defined by $A=\left\{ y \in \mathbb{R}:y = \lim_{n \rightarrow \infty} f(x_{n}), \text {for some sequence} x_{n} \rightarrow +\infty\right\}$ Then the set A is necessarily. $A$ connected set $A$ compact set $A$ singleton set None of the above

Let $f: [0, 1]\rightarrow \mathbb{R}$ be a fixed continuous function such that $f$ is differentiable on $(0, 1)$ and $f(0) = f(1) = 0$. Then the equation $f(x) = f' (x)$ admits. No solution $x \in (0, 1)$ More than one solution $x \in (0, 1)$ Exactly one solution $x \in (0, 1)$ At least one solution $x \in (0, 1)$