Questions and Solutions
Question 1:
Prove that the function \( f(x) = 5x - 3 \) is continuous at \( x = 0 \), at \( x = -3 \) and at \( x = 5 \).
Question 2:
Examine the continuity of the function \( f(x) = 2x^{2} - 1 \) at \( x = 3 \).
Question 3:
Examine the following functions for continuity: (a) \( f(x) = x - 5 \), (b) \( f(x) = \frac{1}{x - 5}, x \neq 5 \), (c) \( f(x) = \frac{x^{2} - 25}{x + 5}, x \neq -5 \), (d) \( f(x) = |x - 5| \).
Question 4:
Prove that the function \( f(x) = x^{n} \) is continuous at \( x = n \), where \( n \) is a positive integer.
Question 5:
Is the function \( f(x) = \begin{cases} x, & \text{if } x \leq 1 \\ 5, & \text{if } x > 1 \end{cases} \) continuous at \( x = 0 \), \( x = 1 \), and \( x = 2 \)?
Question 6:
Find all points of discontinuity of \( f(x) = \begin{cases} 2x + 3, & x \leq 2 \\ 2x - 3, & x > 2 \end{cases} \).
Question 7:
Find all points of discontinuity of \( f(x) = \begin{cases} |x| + 3, & x \leq -3 \\ -2x, & -3 < x < 3 \\ 6x + 2, & x \geq 3 \end{cases} \).
Question 8:
Find all points of discontinuity of \( f(x) = \begin{cases} \frac{|x|}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases} \).
Question 9:
Find all points of discontinuity of \( f(x) = \begin{cases} \frac{x}{|x|}, & x < 0 \\ -1, & x \geq 0 \end{cases} \).
Question 10:
Find all points of discontinuity of \( f(x) = \begin{cases} x + 1, & x \geq 1 \\ x^{2} + 1, & x < 1 \end{cases} \).
Question 11:
Find all points of discontinuity of \( f(x) = \begin{cases} x^{3} - 3, & x \leq 2 \\ x^{2} + 1, & x > 2 \end{cases} \).
Question 12:
Find all points of discontinuity of \( f(x) = \begin{cases} x^{10} - 1, & x \leq 1 \\ x^{2}, & x > 1 \end{cases} \).
Question 13:
Is the function \( f(x) = \begin{cases} x + 5, & x \leq 1 \\ x - 5, & x > 1 \end{cases} \) continuous?
Question 14:
Discuss the continuity of the function \( f(x) = \begin{cases} 3, & 0 \leq x \leq 1 \\ 4, & 1 < x < 3 \\ 5, & 3 \leq x \leq 10 \end{cases} \).
Question 15:
Discuss the continuity of the function \( f(x) = \begin{cases} 2x, & x < 0 \\ 0, & 0 \leq x \leq 1 \\ 4x, & x > 1 \end{cases} \).
Question 16:
Discuss the continuity of the function \( f(x) = \begin{cases} -2, & x \leq -1 \\ 2x, & -1 < x \leq 1 \\ 2, & x > 1 \end{cases} \).
Question 17:
Find the relationship between \( a \) and \( b \) so that the function \( f(x) = \begin{cases} ax + 1, & x \leq 3 \\ bx + 3, & x > 3 \end{cases} \) is continuous at \( x = 3 \).
Question 18:
For what value of \( \lambda \) is the function \( f(x) = \begin{cases} \lambda (x^{2} - 2x), & x \leq 0 \\ 4x + 1, & x > 0 \end{cases} \) continuous at \( x = 0 \)? Also discuss continuity at \( x = 1 \).
Question 19:
Show that the function \( g(x) = x - [x] \) is discontinuous at all integral points, where \( [x] \) denotes the greatest integer less than or equal to \( x \).
Question 20:
Is the function \( f(x) = x^{2} - \sin x + 5 \) continuous at \( x = \pi \)?
Question 21:
(a) Discuss the continuity of the function \( f(x) = \sin x + \cos x \). (b) Discuss the continuity of \( f(x) = \sin x - \cos x \). (c) Discuss the continuity of \( f(x) = \sin x \cdot \cos x \).
Question 22:
Discuss the continuity of the cosine, cosecant, secant, and cotangent functions.
Question 23:
Find all points of discontinuity of \( f(x) = \begin{cases} \frac{\sin x}{x}, & x < 0 \\ x + 1, & x \geq 0 \end{cases} \).
Question 24:
Determine if \( f(x) = \begin{cases} x^{2} \sin \frac{1}{x}, & x \neq 0 \\ 0, & x = 0 \end{cases} \) is a continuous function.
Question 25:
Examine the continuity of \( f(x) = \begin{cases} \sin x - \cos x, & x \neq 0 \\ 1, & x = 0 \end{cases} \).
Question 26:
Find the values of \( k \) so that the function \( f(x) = \begin{cases} \frac{k \cos x}{\pi - 2x}, & x \neq \frac{\pi}{2} \\ 3, & x = \frac{\pi}{2} \end{cases} \) is continuous at \( x = \frac{\pi}{2} \).
Question 27:
Find the values of \( k \) so that the function \( f(x) = \begin{cases} k x^2, & x \le 2 \\ 3, & x > 2 \end{cases} \) is continuous at \( x = 2 \).
Question 28:
Find the values of \( k \) so that the function \( f(x) = \begin{cases} kx + 1, & x \le \pi \\ \cos x, & x > \pi \end{cases} \) is continuous at \( x = \pi \).
Question 29:
Find the values of \( k \) so that the function \( f(x) = \begin{cases} kx + 1, & x \le 5 \\ 3x - 5, & x > 5 \end{cases} \) is continuous at \( x = 5 \).
Question 30:
Find the values of \( a \) and \( b \) such that the function \( f(x) = \begin{cases} 5, & x \le 2 \\ ax + b, & 2 < x < 10 \\ 21, & x \ge 10 \end{cases} \) is continuous.
Question 31:
Show that the function \( f(x) = \cos(x^2) \) is continuous.
Question 32:
Show that the function \( f(x) = | \cos x | \) is continuous.
Question 33:
Examine that \( \sin |x| \) is a continuous function.
Question 34:
Find all the points of discontinuity of \( f(x) = |x| - |x + 1| \).