Chapter 12

Let a function \(f: R \rightarrow R\) satisfy the equation \(f(x+y)=\) \(f(x)+f(y)\) for all \(x, y\). If the function \(f(x)\) is continuous at \(x=0\), then (A) \(f(x)=0\) continuous for all \(x\) (B) \(f(x)\) is continuous for all positive real \(x\) (C) \(f(x)\) is continuous for all \(x\) (D) None of these

The function \(f(x)=[x] \cos \left(\frac{2 x-1}{2}\right) \pi\), where \([.]\) denotes the greatest integer function, is discontinuous at (A) all \(x\) (B) all integer points (C) no \(x\) (D) \(x\) which is not an integer.

The function \(f(x)=[x]^{2}-\left[x^{2}\right](\) where \([x]\) is the greatest integer less than or equal to \(x\) ), is discontinuous at (A) all integers (B) all integers except 0 and 1 (C) all integers except 0 (D) all integers except 1

The function \(f(x)=\frac{1}{u^{2}+u-2}\), where \(u=\frac{1}{x-1}\), is discontinuous at the points (A) \(x=-2,1, \frac{1}{2}\) (B) \(x=\frac{1}{2}, 1,2\) (C) \(x=1,0\) (D) None of these

If \(f(x)=\sum_{k=0}^{n} a_{k}|x-1|^{k}\), where \(a_{i} \in R\), then (A) \(f(x)\) is continuous at \(x=1\) for all \(a_{k} \in R\) (B) \(f(x)\) is differentiable at \(x=1\) for all \(a_{k} \in \mathrm{R}\) (C) \(f(x)\) is differentiable at \(x=1\), provided \(a_{2 k+1}=0\) (D) \(f(x)\) is continuous at \(x=1\), provided \(a_{2 k}=0\)

If \(f(x)=[x] \sin \left(\frac{\pi}{[x+1]}\right)\), where \([.]\) denotes the greatest integer function, then the points of discontinuity of \(f\) in the domain are (A) \(Z\) (B) \(Z \backslash\\{0\\}\) (C) \(R \backslash[-1,0)\) (D) None of these

If \(f\) is differentiable function satisfying \(f(0)=0\) and if \(g(x)=\frac{f(x)}{x}\), then value, that should be assigned to \(g(0)\), so that \(g\) is continuous at ' 0 ' is (A) 1 (B) 0 (C) \(f(0)\) (D) \(f^{\prime}(0)\)

The value of \(f(0)\) so that the function \(f(x)=\frac{\cos ^{-1}\left(1-\\{x\\}^{2}\right) \sin ^{-1}(1-\\{x\\})}{\\{x\\}-\\{x\\}^{3}}, x \neq 0(\\{x\\}\) denotes fractional part of \(x\) ) becomes continuous at \(x=0\) is (A) \(\frac{\pi}{\sqrt{2}}\) (B) \(\frac{\pi}{4}\) (C) \(\frac{\pi}{2}\) (D) None of these

If the function \(f(x)\) defined as \(f(x)=\left\\{\begin{array}{cl}(\sin x+\cos x)^{\operatorname{cosec} x} & , \frac{-\pi}{2}

Let \(f(x)=[n+p \sin x], x \in(0, \pi), n \in Z\) and \(p\) is a prime number, where \([\cdot]\) denotes the greatest integer function. Then, the number of points where \(f(x)\) is not differentiable, are (A) 0 (B) \(2(p-1)\) (C) \(2 p-1\) (D) None of these

The function \(y=f(x)\), defined parametrically as \(x=2 t-|t-1|\) and \(y=2 t^{2}+t|t|\), is (A) continuous and differentiable for \(x \in R\) (B) continuous for \(x \in R\) and differentiable for \(x \in\) \(R-\\{2\\}\) (C) continuous for \(x \in R\) and differentiable for \(x \in\) \(R-\\{-1,2\\}\) (D) None of these

If \(f(x)\) is a continuous function for all real values of \(x\) satisfying \(x^{2}+(f(x)-2) x+2 \sqrt{3}-3-\sqrt{3} f(x)=0, \forall\) \(x \in R\), then the value of \(f(\sqrt{3})\) is (A) \(\sqrt{3}\) (B) \(1-\sqrt{3}\) (C) \(2(1-\sqrt{3})\) (D) \(2(\sqrt{3}-1)\)

The jump of the function at the point of discontinuity i.e., \(x=1\) of the function \(f(x)=\lim _{n \rightarrow \infty} \frac{\log (2+x)-x^{2 n} \sin x}{1+x^{2 n}}\) is (A) \(\sin 1-\log 3\) (B) \(\sin 1+\log 3\) (C) \(-\sin 1+\log 3\) (D) None of these

The function \(f(x)=\left\\{\begin{array}{cll}\frac{x-1}{e^{x-1}}+1 & , & x \neq 1 \\ 0 & , & x=1\end{array}\right.\) (A) is continuous (B) has removable discontinuity (C) has jump discontinuity (D) has infinite discontinuity

If \(f\) is a continuous function from \(R\) to \(R\) and \(f(f(a))=a\) for some \(a \in R\), then the equation \(f(x)=x\) has (A) no solution (B) exactly one solution (C) at most one solution (D) at least one solution

Let \(f\) be a continuous function on \(R\) such that \(f(1 / 4 n)=\left(\sin e^{n}\right) e^{-n^{2}}+\frac{n^{2}}{n^{2}+1}\). Then, the value of \(f(0)\) is (A) 1 (B) \(\frac{1}{2}\) (C) 0 (D) None of these

A function \(f: R \rightarrow R\), where \(R\) is the set of real numbers satisfies the equation \(f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)+f(0)}{3}\) for all \(x, y\) in \(R\). If the function \(f\) is differentiable at \(x=\) 0 , then \(f\) is (A) linear (B) quadratic (C) cubic (D) biquadratic

If \(f(x)=\left\\{\begin{array}{cc}x^{p} \cos \frac{1}{x}, & x \neq 0 \\ 0, & x=0\end{array}\right.\), then at \(x=0, f(x)\) is (A) continuous if \(p>0\) (B) differentiable if \(p>1\) (C) continuous if \(p>1\) (D) differentiable if \(p>0\)

Let \(g(x)=x f(x)\), where \(f(x)=\left\\{\begin{array}{ll}x \sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0\end{array}\right.\). At \(x=0\), (A) \(g\) is differentiable but \(g^{\prime}\) is not continuous (B) \(\mathrm{g}\) is differentiable while \(f\) is not (C) both \(f\) and \(g\) are differentiable (D) \(g\) is differentiable and \(g^{\prime}\) is continuous

The function \(f(x)=\max .\\{(1-x),(1+x), 2\\}\) \(x \in(-\infty, \infty)\), is (A) continuous at all points (B) differentiable at all points (C) differentiable at all points except at \(x=1\) and \(x\) \(=-1 .\) (D) continuous at all points except at \(x=1\) and \(x=-1\), where it is discontinuous.

The function \(f(x)=(x)\), where \((x)\) denotes the smallest integer \(\geq x\), is (A) continuous at integral points (B) continuous at non-integral points (C) discontinuous at integral points (D) discontinuous at non-integral points

Let \(f(x)=\left\\{\begin{array}{cc}\frac{1}{|x|} & |x| \geq 1 \\ a x^{2}+b & |x|<1\end{array}\right.\) If \(f\) is continuous and differentiable at every point, then (A) \(a=\frac{1}{2}\) (B) \(a=-\frac{1}{2}\) (C) \(b=\frac{3}{2}\) (D) \(b=\frac{-3}{2}\)

Let \([x]\) denotes the greatest integer less than or equal to \(x\). If \(f(x)=[x \sin p x]\), then \(f(x)\) is (A) continuous at \(x=0\) (B) continuous in \((-1,0)\) (C) differentiable at \(x=1\) (D) differentiable in \((-1,1)\)

If \(f(x)=\frac{1}{1-x}\), then the points of discontinuity of the function \(f^{3 n}(x)\), where \(f^{n}=\) fof \(\ldots\) of \((n\) times \()\), are (A) \(x=2\) (B) \(x=0\) (C) \(x=1\) (D) continuous everywhere

The function \(F(x)\), defined as \(F(x)=\lim _{n \rightarrow \infty} \frac{f(x)+x^{2 n} g(x)}{1+x^{2 n}}\) shall be continuous everywhere, if (A) \(f(1)=g(1)\) (B) \(f(-1)=g(-1)\) (C) \(f(1)=-g(1)\) (D) \(f(-1)=-g(1)\)

If the function \(f(x)\) defined as \(f(x)=\left\\{\begin{array}{cl}3 & , x=0 \\\ \left(1+\frac{a x+b x^{3}}{x^{2}}\right)^{V_{x}} & , x>0\end{array}\right.\) is continuous at \(x=0\), then (A) \(a=0\) (B) \(b=e^{3}\) (C) \(a=1\) (D) \(b=\ln 3\)

If the function \(f(x)=\frac{\sin 3 x+a \sin 2 x+b \sin x}{x^{5}}, x \neq 0\) is continuous at \(x=0\), then (A) \(a=-4\) (B) \(b=5\) (C) \(a=4\) (D) \(f(0)=1\)

Let \(f^{\prime \prime}(x)\) be continuous at \(x=0\). If \(\lim _{x \rightarrow 0} \frac{2 f(x)-3 a f(2 x)+b f(8 x)}{\sin ^{2} x}\) exists and \(f(0) \neq 0\), \(f^{\prime}(0) \neq 0\), then (A) \(a=\frac{-7}{9}\) (B) \(b=\frac{1}{3}\) (C) \(a=\frac{7}{9}\) (D) \(b=-\frac{1}{3}\)

If \(f(x)=\left\\{\begin{array}{cc}x-3, & x<0 \\ x^{2}-3 x+2, & x \geq 0\end{array}\right.\) and \(g(x)=f(|x|)+|f(x)|\), then \(g(x)\) is (A) continuous in \(R-\\{0\\}\) (B) continuous in \(R\) (C) differentiable in \(R-\\{0,1,2\\}\) (D) differentiable in \(R-\\{1,2\\}\)

Let \(f(x)=x^{3}-x^{2}+x+1\) and \(g(x)=\left\\{\begin{array}{cc}\max \cdot f(t) & 0 \leq t \leq x \text { for } 0 \leq x \leq 1 \\ 3-x & 1

Let \(f(x)=x^{4}-8 x^{3}+22 x^{2}-24 x\) and \(g(x)= \begin{cases}\min . f(t) & x \leq t \leq x+1,-1 \leq x \leq 1 \\ x-10 & x>1\end{cases}\) Then, in the interval \([-1, \infty), g(x)\) is (A) continuous for all \(x\) (B) discontinuous at \(x=1\) (C) differentiable for all \(x\) (D) not differentiable at \(x=1\)

If \(f(x)=[\tan x]+\sqrt{\tan x-[\tan x]}, 0 \leq x<\frac{\pi}{2}\), where [ - ] denotes the greatest integer function, then (A) \(f(x)\) is continuous in \(\left[0, \frac{\pi}{2}\right)\) (B) \(f(x)\) is not continuous at \(x=0\) (C) \(f(x)\) is continuous at \(x=0, \frac{\pi}{4}\) (D) \(f(x)\) has infinite points of discontinuity

Let \(f(x)=g^{\prime}(x) \frac{e^{1 / x}-e^{-1 / x}}{e^{y_{x}}+e^{-1 / x}}\), where \(g^{\prime}\) is the derivative of \(g\) and is a continuous function, then \(\lim _{x \rightarrow 0} f(x)\) exists if (A) \(g(x)\) is a polynomial (B) \(g(x)=x\) (C) \(g(x)=x^{2}\) (D) \(g(x)=x^{3} h(x)\), where \(h(x)\) is a polynomial

If the function \(f(x)\), defined as \(f(x)= \begin{cases}\frac{a(1-x \sin x)+b \cos x+5}{x^{2}}, x<0 \\ 3 & , x=0 \\ \left\\{1+\left(\frac{c x+d x^{3}}{x^{2}}\right)\right\\}^{1 / x} & , x>0\end{cases}\) is continuous at \(x=0\), then (A) \(a=-1\) (B) \(b=-4\) (C) \(c=0\) (D) \(\log _{e}{\underline{\phantom{xx}}}^{3}\)

If \(f(x)=\left\\{\begin{array}{l}3, x<0 \\ 2 x+1, x \geq 0\end{array}\right.\), then (A) both \(f(x)\) and \(f(|x|)\) are differentiable at \(x=0\) (B) \(f(x)\) is differentiable but \(f(|x|)\) is not differentiable at \(x=0\) (C) \(f(|x|)\) is differentiable but \(f(x)\) is not differentiable at \(x=0\) (D) both \(f(x)\) and \(f(|x|)\) are not differentiable at \(x=0\).

Let \(f(x)=\cos x\) and \(g(x)=[x+2]\), where \([.]\) denotes the greatest integer function. Then, \((\text { gof })^{\prime}\left(\frac{\pi}{2}\right)\) is (A) 1 (B) 0 (C) \(-1\) (D) does not exist

The left-hand derivative of \(f(x)=[x] \sin (\pi x)\) at \(x=k\), \(k\) an integer and \([x]=\) greatest integer \(\leq x\), is (A) \((-1)^{k}(k-1) \pi\) (B) \((-1)^{k-1} \cdot(k-1) \pi\) (C) \((-1)^{k} \cdot k \pi\) (D) \((-1)^{k-1} \cdot k \pi\).

The function \(f(x)=\left\\{\begin{array}{l}\frac{x\left(e^{1 / x}-e^{-1 / x}\right)}{e^{1 / x}+e^{-1 / x}}, x \neq 0 \\ 0 \quad, x=0\end{array}\right.\) is (A) continuous everywhere but not differentiable at \(x=0\) (B) continuous and differentiable everywhere (C) not continuous at \(x=0\) (D) None of these

If \(f(x)=\sum_{n=0}^{\infty} \frac{x^{n}}{n !}(\log a)^{n}\), then at \(x=0, f(x)\) (A) has no limit (B) is continuous (C) is continuous but not differentiable (D) is differentiable

Let \(f(x)=\left\\{\begin{array}{cc}\int_{0}^{x}(5+|1-t|) d t, x>2 \\ 5 x+1, x \leq 2\end{array}\right.\), then at \(x=2\) (A) \(f(x)\) is continuous (B) \(f(x)\) is not continuous (C) \(f(x)\) is differentiable (D) \(f(x)\) is not differentiable

The function \(f(x)=\operatorname{are} \tan \frac{1}{x-5}\) has (A) discontinuity of the first kind at \(x=5\) (B) discontinuity of the second kind at \(x=5\) (C) removable discontinuity at \(x=5\) (D) continuous at \(x=5\).

The function \(f(x)=\frac{1-u^{2}}{2+u^{2}}\), where \(u=\tan x\), has (A) discontinuity of the first kind at \(x=n \pi \pm \frac{\pi}{2}\), \(n \in I\) (B) discontinuity of the second kind at \(x=n \pi \pm \frac{\pi}{2}\), \(n \in I\) (C) removable discontinuity at \(x=n \pi \pm \frac{\pi}{2}, n \in I\) (D) continuous at \(x=n \pi \pm \frac{\pi}{2}, n \in I\)

The function \(f(x)=t^{3}\), where \(t=\left\\{\begin{array}{cl}x-1, & x \leq 0 \\\ x+1, & 0

Let \(f(x)=\frac{1}{[\cos x]}\), where \([\cdot]\) denotes the greatest integer function. Then, the function \(f(x)\) has at \(x=\frac{\pi}{2}\) (A) removable discontinuity (B) discontinuity of first kind from left (C) discontinuity of second kind from left (D) None of these

Function Character of discontinuity I. \(f(x)=|2 \sin 2 x|+2\) at \(x=0\) (A) Oscillating discontinuity II. \(f(x)=\left\\{\begin{array}{cc}\tan \frac{\pi x}{2}, & x<1 \\ x-1, & 1 \leq x<2\end{array}\right.\) at \(x=1\) (B) Infinite discontinuity III. \(f(x)=\left\\{\begin{array}{cc}\sin \frac{1}{x}, & x \neq 0 \\ 0, & x=0\end{array}\right.\) at \(x=0\) (C) Removable discontinuity IV. \(f(x)=\frac{|x+2|}{\tan ^{-1}(x+2)}\) at \(x=-2\) (D) Jump discontinuity

Assertion: Let \(f(x+y)=f(x) f(y)\) for all \(x, y\), where \(f(0) \neq 0 .\) If \(f^{\prime}(0)=2\), then \(f(x)=A e^{2 x}\), where \(A\) is a constant. Reason: \(f^{\prime}(x)=f(x)\)

Assertion: Let \(f: R \rightarrow R\) be a function defined by \(f(x)=\max .\left\\{x, x^{3}\right\\} .\) Then, \(f(x)\) is not differentiable at \(x=-1,0,1\) Reason: \(f(x)=\left\\{\begin{array}{l}x, x \leq-1 \\ x^{3},-11\end{array}\right.\)

Assertion: Let \(f: R \rightarrow R\) be any function. Define \(g: R \rightarrow R\) by \(g(x)=|f(x)|\) for all \(x\). Then, \(g\) is continuous if \(f\) is continuous. Reason: Composition of two continuous functions is continuous

\mathrm{\\{} A s s e r t i o n : ~ T h e ~ f u n c t i o n ~ \(f(x)=\lim _{n \rightarrow \infty} \sum_{r=1}^{n} \frac{[2 r x]}{n^{2}}\), where \([\cdot]\) denotes the greatest integer function, is continuous everywhere. Reason: \(f(x)=x, \forall x\)

Assertion: The function \(f(x)=\) \(\lim _{n \rightarrow \infty} \frac{\cos \pi x-x^{2 n} \sin (x-1)}{1+x^{2 n+1}-x^{2 n}}\) is discontinuous at \(x=\pm 1\) Reason: \(f(x)=\left\\{\begin{array}{cl}\frac{\cos \pi x}{1+x}, & |x|<1 \\\ -1+\sin 2, & x=-1 \\ -1, & x=1 \\ \frac{-\sin (x-1)}{x-1}, & |x|>1\end{array}\right.\)