Recent questions tagged continuity

Let $p:\mathbb R\to\mathbb R$ be continuous at $x=2$ and suppose $p(2)=3$. For a real number $\lambda$, define $q:\mathbb R\to\mathbb R$ by $q(y)=\dfrac{y^2-9}{y-3}$ for ...

Let the piecewise function $f(x)$ be defined by $$f(x)=\begin{cases}\dfrac{\sqrt{1+ax}-\sqrt{1-bx}}{x},&x>0\\\\ c,&x=0\\\\ \dfrac{\ln(1+3x)}{\sin x},&x<0\end{cases}$$ For...

Let $$f(x)=\frac{(x-3)(x^2+1)}{|x-3|} \text{ for } x\ne 3$$ If $f$ is extended to be continuous at $x=3$, what should be the value of $f(3)$?$10$ $-10$ $0$ No such value ...

Consider the piecewise function $$f(x)=\begin{cases}\dfrac{e^{Cx}-1+4x}{\sin(2x)},&x<0\\\\ \dfrac{(C^2-1)x^2+7x+3}{2x^2+\sqrt{x^4+1}},&x\ge 0\end{cases}$$where $C$ is a c...

Consider a function $f:(0,1) \rightarrow\{0,1\}$ defined as follows.For a real number $r \in(0,1), f(r)=1$ if the second digit after the decimal point in $r$ is one of th...

Consider the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as follows:\[f(x)=\left\{\begin{array}{lr}c_{1} e^{x}-c_{2} \log _{\mathrm{e}}\left(\frac{1}{x}\right...

Let $m$ and $n$ be the numbers of points at which $f(x) = \max\{x, x^3, x^5, \dots, x^{21}\}$ for $x \in \mathbb{R}$ is not differentiable and not continuous, respectivel...

Let $f, g, h$ be functions from $\mathbb{R}$ to $\mathbb{R}$ such that\[h(f(x)+g(y))=x y\]for all $x, y \in \mathbb{R}$. Show the following:$h$ is surjective.If $f$ is co...

Prove or disprove each of the statements below.Let $f: \mathbb{R}^{2} \longrightarrow \mathbb{R}$ be a continuous function that takes both positive and negative values. T...

Show that the power series $\sum_{n=1}^{\infty} z^{n!}$ represents an analytic function $f(z)$ in the open unit disk $\Delta$ centred at $0.$ Show that $f(z)$ cannot be e...

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be a function given by$$f(x)= \begin{cases}\frac{1-\cos 2 x}{x^2}, & x<0 \\ \alpha, & x=0 \\ \frac{\beta \sqrt{1-\cos x}}{x}, &...

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function which is twice differentiable everywhere and suppose that $f^{\prime}(q)=0$ for every rational number $q.$ Find t...

Let $f:[0,1] \rightarrow \mathbb{R}$ be such that\[f(x)=\left\{\begin{array}{ll}\frac{1}{2^{n}}, & \text { if } x=\frac{p}{2^{n}}, \text { where } p \text { is an odd int...

Suppose $f$ is a real-valued function defined on the set of real numbers. Let$$f(x)=\frac{\sin x}{x} \text { for } x \neq 0, \quad f(0)=a$$where $a$ is a real number. Let...

​​​​​Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function. Note: $\mathbb{R}$ denotes the set of real numbers.\[f(x)=\left\{\begin{array}{cl}-x, & \text { if } x&lt-2...