For a real number $m$, let $f ( m )$ be the number of intersection points of the line passing through the point $( 0,2 )$ with slope $m$ and the curve $y = x ^ { 3 } - 3 x ^ { 2 } + 1$. What is the maximum value of the real number $a$ such that the function $f ( m )$ is continuous on the interval $( - \infty , a )$? [4 points]
(1) $- 3$
(2) $- \frac { 3 } { 4 }$
(3) $\frac { 3 } { 2 }$
(4) $\frac { 15 } { 4 }$
(5) 6

For a constant $a$, define the function $f ( x )$ as $$f ( x ) = \lim _ { n \rightarrow \infty } \frac { ( a - 2 ) x ^ { 2 n + 1 } + 2 x } { 3 x ^ { 2 n } + 1 }$$ What is the sum of all values of $a$ such that $( f \circ f ) ( 1 ) = \frac { 5 } { 4 }$? [4 points]
(1) $\frac { 11 } { 2 }$
(2) $\frac { 13 } { 2 }$
(3) $\frac { 15 } { 2 }$
(4) $\frac { 17 } { 2 }$
(5) $\frac { 19 } { 2 }$

A function $f ( x )$ continuous on the entire set of real numbers satisfies $$\{ f ( x ) \} ^ { 3 } - \{ f ( x ) \} ^ { 2 } - x ^ { 2 } f ( x ) + x ^ { 2 } = 0$$ for all real numbers $x$. When the maximum value of $f ( x )$ is 1 and the minimum value is 0, what is the value of $f \left( - \frac { 4 } { 3 } \right) + f ( 0 ) + f \left( \frac { 1 } { 2 } \right)$? [4 points]
(1) $\frac { 1 } { 2 }$
(2) 1
(3) $\frac { 3 } { 2 }$
(4) 2
(5) $\frac { 5 } { 2 }$

Let $f ( x )$ be a cubic function with positive leading coefficient, and for a real number $t$, let the function $$g ( x ) = \left\{ \begin{array} { r r } - f ( x ) & ( x < t ) \\ f ( x ) & ( x \geq t ) \end{array} \right.$$ be continuous on the set of all real numbers and satisfy the following conditions. (가) For all real numbers $a$, the value of $\lim _ { x \rightarrow a + } \frac { g ( x ) } { x ( x - 2 ) }$ exists. (나) The set of natural numbers $m$ such that $\lim _ { x \rightarrow m + } \frac { g ( x ) } { x ( x - 2 ) }$ is negative is $\left\{ g ( - 1 ) , - \frac { 7 } { 2 } g ( 1 ) \right\}$. Find the value of $g ( - 5 )$. (Given that $g ( - 1 ) \neq - \frac { 7 } { 2 } g ( 1 )$) [4 points]

For all $(s,t) \in [0,1]^2$, $K(s,t) = k_s(t)$ where $$k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ Show that $K$ is continuous on $[0,1] \times [0,1]$.

115- If $f(x) = [x] + [-x]$ and $g(x) = \begin{cases} f(x) & ; \ x \notin \mathbb{Z} \\ f(x) - 1 & ; \ x \in \mathbb{Z} \end{cases}$, then the number of points of discontinuity of $g$ on the interval $[4, -4]$ is which of the following?
(1) $1$ (2) $2$ (3) $3$ (4) zero

118. The function $f(x) = \dfrac{ax^3 - bx^2 + 2}{ax^3 - bx + 2}$ is discontinuous at exactly two points and has exactly two asymptotes parallel to the coordinate axes. What are the values of $a$ and $b$?
(1) $a = 0,\ b = 2$ (2) $a = 8,\ b = 10$
(3) $a = -2,\ b = 0$ (4) $a = -8,\ b = -6$

16. For a specific value of $k$, the function $$f(x) = \begin{cases} |x - [-x]| & x \in [x] \text{ even} \\ x - [x] + k & x \in [x] \text{ odd} \end{cases}$$ is continuous at $x = n$ and $x = -n$. Which case is correct regarding $n$ specifically? $(k, n \in \mathbb{N})$

• [(1)] $n$ even
• [(2)] $n$ odd
• [(3)] $f$ is continuous for all values of $n$.
• [(4)] $f$ is not continuous for any value of $n$.

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For any real number $x$, let $[ x ]$ be the greatest integer $m$ such that $m \leq x$. Then the number of points of discontinuity of the function $g ( x ) = \left[ x ^ { 2 } - 2 \right]$ on the interval $( - 3,3 )$ is
(A) 5
(B) 9
(C) 13
(D) 16 .

27. Let $h ( x ) = \min \{ x ; x 2 \}$, for every real number of $x$. Then :
(A) $h$ is continuous for all $x$
(B) $h$ is differentiable for all $x$
(C) $\mathrm { h } ^ { \prime } ( \mathrm { x } ) = 1$, for all $\mathrm { x } > 1$
(D) $h$ is not differentiable at two values of $x$

15. The function $f ( x ) = [ x ] 2 - [ x 2 ]$ (where [y] is the greatest integer less than or equal to $y$ ), is discontinuous at :
(A) all integers
(B) all integers except 0 and 1
(C) all integers except 0
(D) all integers except 1

14. If $f ( x ) = \min \left\{ 1 , x ^ { 2 } , x ^ { 3 } \right\}$, then
(A) $\mathrm { f } ( \mathrm { x } )$ is continuous $\forall \mathrm { x } \in \mathrm { R }$
(B) $\mathrm { f } ^ { \prime } ( \mathrm { x } ) > 0 , \forall \mathrm { x } > 1$
(C) $\mathrm { f } ( \mathrm { x } )$ is not differentiable but continuous $\forall \mathrm { x } \in \mathrm { R }$
(D) $f ( x )$ is not differentiable for two values of $x$
Sol. (A), (C) $\mathrm { f } ( \mathrm { x } ) = \min . \left\{ 1 , \mathrm { x } ^ { 2 } , \mathrm { x } ^ { 3 } \right\}$ $\Rightarrow \mathrm { f } ( \mathrm { x } ) = \begin{cases} \mathrm { x } ^ { 3 } , & \mathrm { x } \leq 1 \\ 1 , & \mathrm { x } > 1 \end{cases}$ $\Rightarrow \mathrm { f } ( \mathrm { x } )$ is continuous $\forall \mathrm { x } \in \mathrm { R }$ and non-differentiable at $\mathrm { x } = 1$. [Figure]

If $$f ( x ) = \begin{cases} - x - \frac { \pi } { 2 } , & x \leq - \frac { \pi } { 2 } \\ - \cos x , & - \frac { \pi } { 2 } < x \leq 0 \\ x - 1 , & 0 < x \leq 1 \\ \ln x , & x > 1 , \end{cases}$$ then
(A) $f ( x )$ is continuous at $x = -\frac{\pi}{2}$
(B) $f ( x )$ is not differentiable at $x = 0$
(C) $f ( x )$ is differentiable at $x = 1$
(D) $f ( x )$ is differentiable at $x = -\frac{3}{2}$

Let $f_1 : \mathbb{R} \rightarrow \mathbb{R}$, $f_2 : [0,\infty) \rightarrow \mathbb{R}$, $f_3 : \mathbb{R} \rightarrow \mathbb{R}$ and $f_4 : \mathbb{R} \rightarrow [0,\infty)$ be defined by
$$f_1(x) = \begin{cases} |x| & \text{if } x < 0 \\ e^x & \text{if } x \geq 0 \end{cases}$$
$$f_2(x) = x^2;$$
$$f_3(x) = \begin{cases} \sin x & \text{if } x < 0 \\ x & \text{if } x \geq 0 \end{cases}$$
and
$$f_4(x) = \begin{cases} f_2(f_1(x)) & \text{if } x < 0 \\ f_2(f_1(x)) - 1 & \text{if } x \geq 0 \end{cases}$$
List I (functions) P. $f_4$ is Q. $f_3$ is R. $f_2 \circ f_1$ is S. $f_2$ is
List II (properties)
1. onto but not one-one
2. neither continuous nor one-one
3. differentiable but not one-one
4. continuous and one-one
P Q R S
(A) 3142
(B) 1342
(C) 3124
(D) 1324

Let $f : \left[ - \frac { 1 } { 2 } , 2 \right] \rightarrow \mathbb { R }$ and $g : \left[ - \frac { 1 } { 2 } , 2 \right] \rightarrow \mathbb { R }$ be functions defined by $f ( x ) = \left[ x ^ { 2 } - 3 \right]$ and $g ( x ) = | x | f ( x ) + | 4 x - 7 | f ( x )$, where $[ y ]$ denotes the greatest integer less than or equal to $y$ for $y \in \mathbb { R }$. Then
(A) $f$ is discontinuous exactly at three points in $\left[ - \frac { 1 } { 2 } , 2 \right]$
(B) $f$ is discontinuous exactly at four points in $\left[ - \frac { 1 } { 2 } , 2 \right]$
(C) $g$ is NOT differentiable exactly at four points in $\left( - \frac { 1 } { 2 } , 2 \right)$
(D) $g$ is NOT differentiable exactly at five points in $\left( - \frac { 1 } { 2 } , 2 \right)$

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?
[A] $x = -1$
[B] $x = 0$
[C] $x = 1$
[D] $x = 2$

Let the functions $f: (-1,1) \rightarrow \mathbb{R}$ and $g: (-1,1) \rightarrow (-1,1)$ be defined by $$f(x) = |2x - 1| + |2x + 1| \quad \text{and} \quad g(x) = x - [x],$$ where $[x]$ denotes the greatest integer less than or equal to $x$. Let $f \circ g: (-1,1) \rightarrow \mathbb{R}$ be the composite function defined by $(f \circ g)(x) = f(g(x))$. Suppose $c$ is the number of points in the interval $(-1,1)$ at which $f \circ g$ is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which $f \circ g$ is NOT differentiable. Then the value of $c + d$ is $\_\_\_\_$

For any real number $t$, let $\lfloor t \rfloor$ be the largest integer less than or equal to $t$. Then the number of points of discontinuity of the function $x \mapsto \lfloor x^2 - 3 \rfloor$ for $x \in (-\infty, 0)$ is ____.
Let $f: [-1, 3] \to \mathbb{R}$ be defined as $$f(x) = \begin{cases} |x| + [x], & -1 \leq x < 1 \\ x + |x|, & 1 \leq x < 2 \\ x + [x], & 2 \leq x \leq 3 \end{cases}$$ where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points of discontinuity of $f$ in the interval $(-1, 3)$ is ____.

Let $f ( x ) = \left\{ \begin{array} { c c } \max \left( | x | , x ^ { 2 } \right) , & | x | \leq 2 \\ 8 - 2 | x | , & 2 < | x | \leq 4 \end{array} \right.$. Let $S$ be the set of points in the interval $( - 4,4 )$ at which $f$ is not differentiable. Then $S$
(1) equals $\{ - 2 , - 1,0,1,2 \}$
(2) equals $\{ - 2,2 \}$
(3) is an empty set
(4) equal $\{ - 2 , - 1,1,2 \}$

Let $f : [ - 1,3 ] \rightarrow \mathrm { R }$ be defined as $$f ( x ) = \begin{cases} |x| + [x], & -1 \leq x < 1 \\ x + |x|, & 1 \leq x < 2 \\ x + [x], & 2 \leq x \leq 3 \end{cases}$$ where $[t]$ denotes the greatest integer less than or equal to $t$. Then, $f$ is discontinuous at:
(1) Only one point
(2) Only two points
(3) Four or more points
(4) Only three points

If $f ( x ) = [ x ] - \left[ \frac { x } { 4 } \right] , x \in R$, where $[ x ]$ denotes the greatest integer function, then:
(1) $\lim _ { x \rightarrow 4 + } f ( x )$ exists but $\lim _ { x \rightarrow 4 - } f ( x )$ does not exist
(2) $f$ is continuous at $x = 4$
(3) $\lim _ { x \rightarrow 4 - } f ( x )$ exists but $\lim _ { x \rightarrow 4 + } f ( x )$ does not exist
(4) Both $\lim _ { x \rightarrow 4 - } f ( x )$ and $\lim _ { x \rightarrow 4 + } f ( x )$ exist but are not equal

Let $[ t ]$ denote the greatest integer $\leq t$ and $\lim _ { x \rightarrow 0 } x \left[ \frac { 4 } { x } \right] = A$. Then the function, $f ( x ) = \left[ x ^ { 2 } \right] \sin ( \pi x )$ is discontinuous, when $x$ is equal to:
(1) $\sqrt { A + 1 }$
(2) $\sqrt { A + 5 }$
(3) $\sqrt { A + 21 }$
(4) $\sqrt { A }$

A function $f$ is defined on $[ - 3,3 ]$ as $$f ( x ) = \left\{ \begin{array} { c } \min \left\{ | x | , 2 - x ^ { 2 } \right\} , - 2 \leq x \leq 2 \\ { [ | x | ] , 2 < | x | \leq 3 } \end{array} \right.$$ where $[ x ]$ denotes the greatest integer $\leq x$. The number of points, where $f$ is not differentiable in $( - 3,3 )$ is $\underline{\hspace{1cm}}$.

Let a function $g : [ 0,4 ] \rightarrow R$ be defined as $g ( x ) = \left\{ \begin{array} { c c } \max \left\{ t ^ { 3 } - 6 t ^ { 2 } + 9 t - 3 \right\} , & 0 \leq x \leq 3 \\ 0 \leq t \leq x & \\ 4 - x, & 3 < x \leq 4 \end{array} \right.$ then the number of points in the interval $( 0,4 )$ where $g ( x )$ is NOT differentiable, is $\underline{\hspace{1cm}}$.

Let $f ( x ) = \begin{cases} \frac { \sin ( x - [ x ] ) } { x - [ x ] } , & x \in ( - 2 , - 1 ) \\ \max ( 2 x , 3 [ | x | ] ) , & | x | < 1 \\ 1 , & \text { otherwise } \end{cases}$ where $[ t ]$ denotes greatest integer $\leq t$. If $m$ is the number of points where $f$ is not continuous and $n$ is the number of points where $f$ is not differentiable, the ordered pair $( m , n )$ is:
(1) $( 3,3 )$
(2) $( 2,4 )$
(3) $( 2,3 )$
(4) $( 3,4 )$