List of top Real Analysis Questions on Functions of One Real Variable

Let f(x) = \(\sqrt[3]{x}\) for x ∈ (0, ∞), and θ(h) be a function such that
f(3 + h) − f(3) = hf′ (3 + θ(h)h)
for all h ∈ (−1, 1). Then \(\lim\limits_{h→0} θ(h)\) is equal to _________. (rounded off to two decimal places)

The global minimum value of
f(x) = |x - 1| + |x - 2|²
on \(\R\) is equal to _________. (rounded off to two decimal places)

Consider the system of linear equations\[ \begin{cases}x + y + t = 4, \\2x - 4t = 7, \\x + y + z = 5, \\x - 3y - z - 10t = \lambda,\end{cases}\]where \( x, y, z, t \) are variables and \( \lambda \) is a constant. Then which one of the following is true?

For t ∈ \(\R\), let [𝑡] denote the greatest integer less than or equal to t.
Let D = {(x, y) ∈ \(\R^2\) ∶ x2 + y2 < 4}. Let f : D → \(\R\) and g : D → \(\R\) be defined by f(0, 0) = g(0, 0) = 0 and
\(f(x,y)=[x^2+y^2]\frac{x^2y^2}{x^4+y^4},\ \ \ g(x,y)=[y^2]\frac{xy}{x^2+y^2}\)
for (x, y) ≠ (0, 0). Let E be the set of points of D at which both f and g are discontinuous. The number of elements in the set E is _________.

Let V be the real vector space consisting of all polynomials in one variable with real coefficients and having degree at most 5, together with the zero polynomial. Let T: V→ \(\R\) be the linear map defined by 7(1) = 1 and T(x(x-1)... (x-k+1)) = 1 for 1 ≤ k ≤ 5.
Then which of the following is/are true?

Define f: [0,1] → [0,1] by
\(f(x) = \begin{cases} 1 & \text{if } x =0 \\ \frac{1}{n} & \text{if } x=\frac{m}{n}\ \text{for some}\ m, n \isin \N\ \text{with}\ m\le n\ \text{and gcd(m, n)} = 1, \\ 0 & \text{if} x\isin[0,1]\ \text{is irrational}\end{cases}\)
and define g: [0,1]→ [0,1] by
\(g(n) = \begin{cases} 0 & \text{if } x=0 \\ 1 & \text{if } x\isin(0,1]. \end{cases}\)
Then which of the following is/are true?

A real-valued function y(x) defined on \(\R\) is said to be periodic if there exists a real number \(T\gt0\) such that y(x + T) = y(x) for all \(x\isin\R\). Consider the differential equation
\(\frac{d^2y}{dx^2}+4y=\sin(ax), x\isin\R, \quad (*)\)
where \(a\isin\R\) is a constant.
Then which of the following is/are true?

Let \( S = \left\{ \frac{1}{n} : n \in \mathbb{N} \right\} \) and \( f : S \to \mathbb{R} \) be defined by \( f(x) = \frac{1}{x}. \) Then \[ \max \left\{ \delta : |x - \tfrac{1}{3}| < \delta \Rightarrow |f(x) - f(\tfrac{1}{3})| < 1 \right\} \] is ............. (rounded off to two decimal places).

Let \( a, b \in \mathbb{R} \) and \( a < b. \) Which of the following statement(s) is/are true?

Let \( a, b, c \in \mathbb{R} \) such that \( a < b < c. \) Which of the following is/are true for any continuous function \( f : \mathbb{R} \to \mathbb{R} \) satisfying \( f(a) = b, f(b) = c \) and \( f(c) = a? \)

Let \( f : [0, 1] \to \mathbb{R} \) be a continuous function such that \( f\left(\dfrac{1}{2}\right) = -\dfrac{1}{2} \) and \[ |f(x) - f(y) - (x - y)| \le \sin(|x - y|^2) \] for all \( x, y \in [0, 1]. \) Then \( \int_0^1 f(x) \, dx \) is

Let \( a \in \mathbb{R} \). If \( f(x) = \begin{cases} (x + a)^2, & x \leq 0 \\ (x + a)^3, & x > 0 \end{cases} \), then

Let \( f(x) = 2x^3 - 9x^2 + 7. \) Which of the following is true?

Let \( f : \mathbb{R} \to \mathbb{R} \) be such that \( f, f', f'' \) are continuous with \( f > 0, f' > 0, f'' > 0. \) Then \( \displaystyle \lim_{x \to -\infty} \frac{f(x) + f'(x)}{2} \) is ............

Let \( f : \mathbb{R} \setminus \{0\} \to \mathbb{R} \) be defined by \( f(x) = x + \dfrac{1}{x^3} \). On which of the following interval(s) is \( f \) one-to-one?

For \( x > -\dfrac{1}{2} \), let \( f_1(x) = \dfrac{2x}{1+2x} \), \( f_2(x) = \log_e(1 + 2x) \) and \( f_3(x) = 2x \). Then which one of the following is TRUE?

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function and let \( J \) be a bounded open interval in \( \mathbb{R} \). Define
\[ W(f, J) = \sup \{ f(x) | x \in J \} - \inf \{ f(x) | x \in J \} \] Which one of the following is FALSE?

For \( x \in \mathbb{R} \), let \( f(x) = \begin{cases} x^3 \sin \left( \frac{1}{x} \right), & x \neq 0 \\ 0, & x = 0 \end{cases} \) . Then which one of the following is FALSE?

Let \( f : \mathbb{R} \to \mathbb{R} \) be such that \( f'' \) is continuous on \( \mathbb{R} \) and \( f(0) = 1 \), \( f'(0) = 0 \), and \( f''(0) = -1 \). \[ \text{Then} \,\, \lim_{x \to \infty} \left( f \left( \sqrt{\frac{2}{x}} \right) \right)^x \text{ is ............} \text{ (correct up to three decimal places)}. \]