1. Prove that the function \(f(x)=5x–3\) is continuous at \(x=0\), at \(x=–3\) and at \(x=5\).

2. Examine the continuity of the function \(f(x)=2x^2–1\) at \(x=3\).

3. Examine the following functions for continuity.
(i) \(f(x)=x-5\)
(ii) \(f(x)=\frac{1}{x-5}\),\(x≠5\)
(iii) \(f(x)=\frac{x^2-25}{x+5}\),\(x≠5\)
(iv) \(f(x)=|x-5|\)

4. Prove that the function \(f(x)=x^n\) is continuous at \(x=n\), where n is a positive integer.

5. Is the function f defined by \(f(x)=\begin{cases}x, & \text{if} & x≤1\\ 5, & \text{if} & x>1 \end{cases}\) continuous at \(x=0\)? At \(x=1\)? At \(x=2\)?

Find all points of discontinuity of f, where f is defined by:

6. \(f(x)=\begin{cases}2x+3, & \text{if} & x≤2\\2x-3,  & \text{if} & x>2 \end{cases}\)

7. \(f(x)=\begin{cases}|x|+3, & \text{if} & x≤-3\\-2x,  & \text{if} & -3<x<3\\ 6x+2, & \text{if} & x\ge 3  \end{cases}\)

8. \(f(x)=\begin{cases}\frac{|x|}{x}, & \text{if} & x\ne 0\\0,  & \text{if} & x=0 \end{cases}\)

9. \(f(x)=\begin{cases}\frac{x}{|x|}, & \text{if} & x<0\\-1,  & \text{if} & x\ge 0 \end{cases}\)

10. \(f(x)=\begin{cases}x+1, & \text{if} & x\ge 1\\x^2+1,  & \text{if} & x<1 \end{cases}\)

11. \(f(x)=\begin{cases}x^3-3, & \text{if} & x\le 2\\x^2+1,  & \text{if} & x>2 \end{cases}\)

12. \(f(x)=\begin{cases}x^{10}-1, & \text{if} & x\le 1\\x^2,  & \text{if} & x>1 \end{cases}\)

13. Is the function defined by \(f(x)=\begin{cases}x+5, & \text{if} & x\le 1\\x-5,  & \text{if} & x>1 \end{cases}\) a continuous function ?

Discuss the continuity of the function f, where f is defined by:
14. \(f(x)=\begin{cases}3, & \text{if} & 0≤x≤1\\4,  & \text{if} & 1<x<3\\ 5, & \text{if} & 3≤x≤10\end{cases}\)

15. \(f(x)=\begin{cases}2x, & \text{if} & x<0\\0,  & \text{if} & 0≤x≤1\\ 4x, & \text{if} & x>1\end{cases}\)

16. \(f(x)=\begin{cases}-2, & \text{if} & x≤-1\\2x,  & \text{if} & -1<x≤1\\ 2, & \text{if} & x>1\end{cases}\)

17. Find the relationship between a and b so that the function f defined by:
\(f(x)=\begin{cases}ax+1, & \text{if} & x\le 3\\bx+3,  & \text{if} & x>3 \end{cases}\) is continuous at x=3.

18. For what value of λ is the function defined by \(f(x)=\begin{cases}λ(x^2-2x), & \text{if} & x\le 0\\4x+1,  & \text{if} & x>0 \end{cases}\)  continuous at x=0? What about continuity at x=1?

19. Show that the function defined by \(g(x)=x–[x]\) is discontinuous at all integral points. Here [x] denotes the greatest integer less than or equal to x.

20. Is the function defined by \(f(x)=x^2 –sinx+5\) continuous at \(x=π\)?

21. Discuss the continuity of the following functions:
(a) \(f(x)=sinx+cosx\)
(b) \(f(x)=sinx–cosx\)
(c) \(f(x)=sinx.cosx\)

22. Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

23. Find all points of discontinuity of f, where
\(f(x)=\begin{cases}\frac{sinx}{x}, & \text{if} & x< 0\\x+1,  & \text{if} & x\ge 0 \end{cases}\)

24. Determine if f defined by \(f(x)=\begin{cases}x^2 sin(\frac{1}{x}), & \text{if} & x\ne 0\\0,  & \text{if} & x=0 \end{cases}\) is a continuous function?

25. Examine the continuity of f, where f is defined by \(f(x)=\begin{cases}sinx-cosx, & \text{if} & x\ne 0\\-1,  & \text{if} & x=0 \end{cases}\).

Find the values of k so that the function f is continuous at the indicated point in Exercises 26 to 29.

26. \(f(x)=\begin{cases}\frac{k cosx}{\pi – 2x}, & \text{if} & x\ne \frac{\pi}{2}\\3,  & \text{if} & x=\frac{\pi}{2} \end{cases}\) at \(x=\frac{\pi}{2}\).

27. \(f(x)=\begin{cases}kx^2, & \text{if} & x\le 2 \\3,  & \text{if} & x>2 \end{cases}\) at \(x=2\).

28. \(f(x)=\begin{cases}kx+1, & \text{if} & x\le \pi \\cosx,  & \text{if} & x>\pi \end{cases}\) at \(x=\pi\).

29. \(f(x)=\begin{cases}kx+1, & \text{if} & x\le 5 \\3x-5,  & \text{if} & x>5 \end{cases}\) at \(x=5\).

30. Find the values of a and b such that the function defined by: \(f(x)=\begin{cases}5, & \text{if} & x≤2\\ax+b,  & \text{if} & 2<x<10\\21, & \text{if} & x\ge 10\end{cases}\) is a continuous function.

31. Show that the function defined by \(f(x)=cos(x^2)\) is a continuous function.

32. Show that the function defined by \(f(x)=|cosx|\) is a continuous function.

33. Examine that \(sin|x|\) is a continuous function.

34. Find all the points of discontinuity of f defined by \(f(x)=|x|–|x+1|\).