Recent questions tagged functions

Which of the following sequences of functions $\left \{ f_{n} \right \}_{n=1}^{\infty }$ converges uniformly ?$f_{n}\left ( x \right )=x^{n}\:on \: \left [ 0,1 \right ]$$...

Let $f$ be a continuous function on $\mathbb{R}$ satisfying the relation$$f\left ( f\left ( f\left ( x \right ) \right ) \right )=x \:for\:all \:x \in \mathbb{R}.$$Prove ...

Prove or disprove the following: let $f:X\rightarrow X$ be a continuous function from a complete metric space $\left ( X,d \right )$ into itself such that $d\left ( f\lef...

Show that there does not exist any continuous function $f:\mathbb{R} \rightarrow \mathbb{R}$ that takes every value exactly twice.

True/False Question :Let $f:\left [ 0,1 \right ]\rightarrow \mathbb{R}$ be a continuous function such that $f\left ( x \right )\geq x^{3}$ for all $x \in \left [ 0,1 \rig...

True/False Question :There exists a function $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying,$f\left ( -1 \right )=-1, f(1)=1$ and $\left | f\left ( x \right )-f\left ( ...

True/False Question :If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and bijective, then $f^{-1}$ is also differentiable.

True/False Question :Given a continuous function $f:\mathbb{Q}\rightarrow \mathbb{Q}$, there exists a continuous function $g:\mathbb{R}\rightarrow \mathbb{R}$ such that t...

True/False Question :There is a continuous onto function $f:S^{2}\rightarrow S^{1}$ from the unit sphere in $\mathbb{R}^{3}$ to the unit sphere in $\mathbb{R}^{2}$, where...

True/False Question :Let $P$ be a monic, non-zero, polynomial of even degree, and $K>0$. Then the function $P\left ( x \right )-K e^{x}$ has a real zero.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be defined as follows:$$f\left ( x \right )=\left\{\begin{matrix} 1, & if &x=0 \\ 0,&if &x \in \mathbb{R}\setminus \mathbb{Q}, \:...

Let $A$ be the set of all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following two properties:$f$ has derivatives of all orders, andfor all $x,y \in ...

Consider the following four sets of maps $f:\mathbb{Z}\rightarrow \mathbb{Q}$:$\{f:\mathbb{Z}\rightarrow \mathbb{Q} \mid f$ is bijective and increasing$\}$,$\{f:\mathbb{Z...

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous. Then which of the following statements implies that $f\left ( 0 \right )=0$?$\underset{n\rightarrow \infty }{lim}\...

True/False Question ;Let $f$ be a twice differentiable function on $\mathbb{R}$ such that both $f$ and ${f}''$ are strictly positive on $\mathbb{R}$. Then $\underset{x\r...

True/False Question :There exists a continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f\left ( \mathbb{Q} \right )\subseteq \mathbb{R}-\mathbb{Q}$ and $...

True/False Question :There exists an injective ring homomorphism from the product ring $\mathbb{R}\times \mathbb{R}$ into $C\left ( \mathbb{R} \right )$, where $C\left ( ...

True/False Question :There exists a uniformly continuous function $f:\left ( 0,\infty \right )\rightarrow \left ( 0,\infty \right )$ such that $$\sum_{n=1 }^{\infty }\fra...

Define a function:$$f\left ( x \right )=\left\{\begin{matrix} x +x^{2} cos\left ( \frac{\pi}{x} \right ), & x\neq 0\\ 0,& x=0. \end{matrix}\right.$$Consider the following...

Consider differentiable functions $f:\mathbb{R} \rightarrow \mathbb{R}$ with the property that for all $a,b \in \mathbb{R}$ we have:$$f\left ( b \right )-f\left ( a \righ...

Let $f:\left ( 0,\infty \right )\rightarrow \mathbb{R}$ be defined by $f\left ( x \right )=\frac{sin\left (x ^{3} \right )}{x}$ . Then $f$ isbounded and uniformly continu...

Let $\left \{ f_{n} \right \}_{n=1}^{\infty}$ be a sequence of functions from $\mathbb{R}$ to $\mathbb{R}$, defined by$$f_{n}\left ( x \right )=\frac{1}{n}\:exp\left ( -n...

Consider functions $f:\mathbb{R}\rightarrow \mathbb{R}$ with the property that $\left | f\left ( x \right )-f\left ( y \right ) \right |\leq 4321\left | x-y \right |$ fo...

Let $\left \{ f_{n} \right \}_{n=1}^{\infty }$ be a sequence of functions from $\mathbb{R}$ to $\mathbb{R}$, defined by$$f_{n}\left ( x \right )=\frac{\sqrt{1+\left ( nx^...

Let $C^{\infty }\left ( 0,1 \right )$ stand for the set of all real-valued functions on $\left ( 0,1 \right )$ that have derivatives of all orders. Then the map $C^{\inft...

True/False Question :There exists no monotone function $f:\mathbb{R}\rightarrow \mathbb{R}$ which is discontinuous at every rational number.

True/False Question :Let $C\left ( \left [ 0,1 \right ] \right )$ denote the set of continuous real valued functions on $\left [ 0,1 \right ]$, and $\mathbb{R}^{\mathbb{N...