Continuity and Differentiability — JEE Advanced PYQs

Let $\mathbb{R}$ denote the set of all real numbers. Let $f:\mathbb{R}\to \mathbb{R}$ be an arbitrary function and let $g:\mathbb{R}\to \mathbb{R}$ be the function defined by

Then which of the follo …

Consider the function $f:\left(-\frac{\pi }{2},\frac{\pi }{2}\right)\to (-\infty ,\infty )$ defined by

where [x sin x] is the greatest integer less than or equal to x …

For a real number $\alpha$, let $[\alpha ]$ denote the greatest integer less than or equal to $\alpha$. For a finite set S , let $|\mathrm{S}|$ denote the number of elements in the set S .

Consider the functions $f:(-3,3)\to (-\infty ,\infty )$ and …

Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let $[x]$ denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.
Match each entry in List-I to the correct entry in List-II and choose the correct option.

| | L …

Let $x_0$ be the real number such that $e^{x_0}+x_0=0$. For a given real number $\alpha$, define

for all real numbers $x$.
Then which one of the following statements is TRUE?

Let $\mathbb{R}$ denote the set of all real numbers. Let $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to (0,4)$ be functions defined by

Define the composite function …

Let $f:\mathbb{R}\to \mathbb{R}$ and $\mathrm{g}:\mathbb{R}\to \mathbb{R}$ be functions defined by …

Let $\mathrm{f}:(0,1)\to \mathbb{R}$ be the function defined as $\mathrm{f}(\mathrm{x})=[4\mathrm{x}]{\left(\mathrm{x}-\frac{1}{4}\right)}^2\left(\mathrm{x}-\frac{1}{2}\right)$, where [x] denotes the greatest integer less than or equal to x . Then which of the following …

Let $f:\left[-\frac{\pi }{2},\frac{\pi }{2}\right]\to \mathbb{R}$ be a continuous function such that

Then which of the following statements is (are) TRUE?

Let the function $f:\mathbb{R}\to \mathbb{R}$ be defined by $f(x)=x^3-x^2+(x-1)\sin x$ and let $g:\mathbb{R}\to \mathbb{R}$ be an arbitrary function. Let $fg:\mathbb{R}\to \mathbb{R}$ be the product function defined by $(fg)(x)=f(x)g(x)$. Then …

Let the functions $f:(-1,1)\to \mathbb{R}$ and $g:(-1,1)\to (-1,1)$ be defined by

where $[x]$ denotes the greatest integer less than or equal to $x$. Let $f\circ g:(-1,1)\to \mathbb{R}$ be the co …

Let $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ be functions satisfying $f(x+y)=f(x)+f(y)+f(x)f(y)$ and $f(x)=xg(x)$ for all $x,y\in \mathbb{R}$. If ${\lim }_{x\to 0}g(x)=1$, then which of the following statements is/are …

Let $\mathrm{F}:\mathbb{R}\to \mathbb{R}$ be a function. We say that f has
PROPERTY 1 if ${\lim }_{\mathrm{h}\to 0}\frac{\mathrm{f}(\mathrm{h})-\mathrm{f}(0)}{\sqrt{|\mathrm{h}|}}$ exists and is finite, and
PROPERTY 2 if …

Let $f_1:\mathbb{R}\to \mathbb{R},f_2:\left(-\frac{\pi }{2},\frac{\pi }{2}\right)\to \mathbb{R},f_3:\left(-1,e^{\frac{\pi }{2}}-2\right)\to \mathbb{R}$ and $f_4:\mathbb{R}\to \mathbb{R}$ be functions defined by
(i) …

Let $[x]$ be the greatest integer less than or equals to $x$. Then, at which of the following point(s) the function $f(x)=x\cos (\pi (x+[x]))$ is discontinuous?

Let $f:\mathbb{R}\to (0,1)$ be a continuous function. Then, which of the following function(s) has(have) the value zero at some point in the interval (0, 1)?

Let $f:\mathbb{R}\to \mathbb{R},g:\mathbb{R}\to \mathbb{R}$ and $h:\mathbb{R}\to \mathbb{R}$ be differentiable functions such that $f(x)=x^3+3x+2$, $g(f(x))=x$ and $h(g(g(x)))=x$ for all $x\in \mathbb{R}$. Then

Let $a,b\in \mathbb{R}$ and $f:\mathbb{R}\to \mathbb{R}$ be defined by $f(x)=a\cos \left(\left|x^3-x\right|\right)+b|x|\sin \left(\left|x^3+x\right|\right)$. Then $f$ is

Let $f:\left[-\frac{1}{2},2\right]\to \mathbb{R}$ and $g:\left[-\frac{1}{2},2\right]\to \mathbb{R}$ be functions defined by $f(x)=\left[x^2-3\right]$ and $g(x)=|x|f(x)+|4x-7|f(x)$, where $[y]$ denotes the greatest integer less than or equal to …

Let $g:\mathbb{R}\to \mathbb{R}$ be a differentiable function with $g(0)=0,g^{\prime }(0)=0$ and $g^{\prime }(1)\ne 0$. Let $f(x)=\{\begin{array}{cc}\frac{x}{|x|}g(x),& x\ne 0\\ 0,& x=0\end{array}$
and $h(x)=e^{|x|}$ for all $x\in \mathbb{R}$. Let …

For every pair of continuous functions $f,g:[0,1]\to \mathbb{R}$ such that

the correct statement(s) is(are) :

Let $f:[a,b]\to [1,\infty )$ be a continuous function and let $g:\mathbb{R}\to \mathbb{R}$ be defined as …

Let $f:\mathbb{R}\to \mathbb{R}$ and $g:\mathbb{R}\to \mathbb{R}$ be respectively given by $f(x)=|x|+1$ and $g(x)=x^2+1$. Define $h:\mathbb{R}\to \mathbb{R}$ by …

Let $f_1:\mathbb{R}\to \mathbb{R},f_2:[0,\infty )\to \mathbb{R},f_3:\mathbb{R}\to \mathbb{R}$ and $f_4:\mathbb{R}\to [0,\infty )$ be defined by …

Let $f(x)=(1-x{)}^2{\sin }^{2}x+x^2$ for all $x\in \mathbb{R}$, and let $g(x)={\int }_1^x\left(\frac{2(t-1)}{t+1}-\ln t\right)f(t)dt$ for all $x\in (1,\infty )$.

Consider the statements :
$\mathbf{P}$ : There exists some $x\in \mathbb{R}$ such that …

For every integer $n$, let $a_n$ and $b_n$ be real numbers. Let function $f:\mathbb{R}\to \mathbb{R}$ be given by …

Let $f(x)=\{\begin{array}{rl}{x}^2\left|\cos \frac{\pi }{x}\right|,& x\ne 0\\ 0,& x=0\end{array},x\in \mathbb{R}$, then $f$ is

If

then

In the following $[x]$ denotes the greatest integer less than or equal to $x$.
Match the functions in Column I with the properties in Column II and indicate your answer by darkening the appropriate bubbles in the 4 × 4 matrix given in the ORS.
| Column I | Column II || --- …

If $f(x)=\min \{1,x^2,x^3\}$, then