Continuity and Differentiability

If the function $f(x)=\left\{\begin{array}{ll}a+b x, & x \lt 1 \\ 5, & x=1 \\ b-a x, & x \gt 1\end{array}\right.$ is continuous, then what is the value of $(a+b)$ ?

A function is defined in $(0, \infty)$ by $f(x)=\left\{\begin{array}{ccc}1-x^2 & \text { for } & 0 \lt x \leq 1 \\\text { In } & x & \text { for } 1 \lt x \leq 2 \\\ln 2-1+0.5 x & \text { for } & 2 \lt x \lt…

Passage: Let $f(x)=\tan(x^2)$ and $g(x)=x|x|$ for $|x|Question: If $$q(x)=f\circ$ g(x)$, then which of the following statements is/are correct ? I. $q(x)$ is continuous at $x=0$. II. $q(x)$ is differentiable at $x=0$.…

Let $g$ be the greatest integer function. Then the function $f(\mathrm{x})=(\mathrm{g}(\mathrm{x}))^2-\mathrm{g}(\mathrm{x})$ is discontinuous at

If $f(x)$ is differentiable at $x=a$, then consider the following statements : I. $f(x)$ is continuous at $x=a$ II. $\lim\limits_{x\to a}f(x)=f(a)$ Which of the statements given above is/are correct ?

For what value of $k$ is the function $f(x)=\left\{\begin{array}{ccc} 2 x+\frac{1}{4} & , & x \lt 0 \\ k & , & x=0 \text { continuous? } \\ \left(x+\frac{1}{2}\right)^2 & , & x \gt 0 \end{array}\right.$

Passage: Let $f(x)=\tan(x^2)$ and $g(x)=x|x|$ for $|x|Question: If $p(x)=f(x)g(x)$, then which of the following statements is/are correct ? I. $p(x)$ is continuous at $x=0$. II. $p(x)$ is differentiable at $x=0$. Select…

Consider the function $f(x)=\left\{\begin{array}{l} \frac{\sin 2 x}{5 x} \text { if } x \neq 0 \\ \frac{2}{15} \text { if } x=0 \end{array}\right.$ Which one of the following is correct in respect of the function?

Paragraph: Let $f(x)=[x]^2-\left[x^2\right]$. Question: Consider the following statements: A. $f(x)$ is continuous at $x=0$. B. $f(x)$ is continuous at $x=1$ Which of the statements given above is/are correct?

Passage: Let $f(x)=\begin{cases} ax(x-1), & x3 \end{cases}$ Given that $f(x)$ is continuous for all $x$ but not differentiable at $x=1$. Further $f'(x)$ is continuous at $x=3$. Question: What is the value of $q$ ?

A function is defined as follows: $f(\mathrm{x})=\left\{\begin{array}{cc} -\frac{\mathrm{x}}{\sqrt{\mathrm{x}^2}}, & \mathrm{x} \neq 0 \\ 0, & \mathrm{x}=0 \end{array}\right.$ Which one of the following is correct in…

Paragraph: Let $\mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}-2, & -3 \leq \mathrm{x} \leq 0 \\ \mathrm{x}-2, & 0 \lt \mathrm{x} \leq 3\end{array}\right.$ and…