Q5. Examine the continuity of (a) f(x) = x − 5; (b) f(x) = 1/(x − 5), x ≠ 5; (c) f(x) = (x2 − 25)/(x + 5), x ≠ −5; (d) f(x) = |x − 5| (NCERT Exercise 5.1, Q3, page 116).
(a) is a polynomial and (b), (c) are rational functions, continuous on their domains; (d) is |·| composed with x − 5. All four are continuous functions. (In (c), −5 is not in the domain, so it is not a point of discontinuity.)
Q6. Is f(x) = x3 − 3 for x ≤ 2 and x2 + 1 for x > 2 continuous? (NCERT Exercise 5.1, Q11, page 116)
At 2, f(2) and the left-hand limit are 8 − 3 = 5, and the right-hand limit is 4 + 1 = 5. Elsewhere each piece is a polynomial. So there is no point of discontinuity.
Q7. Find the points of discontinuity of f(x) = x/|x| for x < 0 and −1 for x ≥ 0 (NCERT Exercise 5.1, Q9, page 116).
For x < 0, |x| = −x, so x/|x| = x/(−x) = −1. So f(x) = −1 for every x: a constant function, with no point of discontinuity.
Q8. Discuss the continuity of f(x) = x2 sin (1/x) for x ≠ 0 and f(0) = 0 (NCERT Exercise 5.1, Q24, page 117).
For x ≠ 0 it is a product and composite of continuous functions. At 0: −x2 ≤ x2 sin (1/x) ≤ x2, and both bounds tend to 0, so the limit is 0 = f(0). So f is continuous for all real x.
Q9. Find all points of discontinuity of f(x) = sin x/x for x < 0 and x + 1 for x ≥ 0 (NCERT Exercise 5.1, Q23, page 117).
At 0, the left-hand limit is lim sin x/x = 1 and the right-hand limit and f(0) are 0 + 1 = 1. Elsewhere the pieces are continuous. There is no point of discontinuity.
Q10. Discuss the continuity of the cosine, cosecant, secant and cotangent functions (NCERT Exercise 5.1, Q22, page 117).
Each is a quotient of continuous functions, continuous wherever its denominator is not 0: cosec x = 1/sin x for x ≠ nπ, sec x = 1/cos x for x ≠ (2n + 1)π/2, and cot x = cos x/sin x for x ≠ nπ, with n an integer. cos x itself is continuous for every real x.
Q11. Show that |cos x| and sin |x| are continuous functions (NCERT Exercise 5.1, Q32 and Q33, page 118).
|cos x| is |·| composed with cos, and sin |x| is sin composed with |·|. Both inner and outer functions are continuous everywhere, so by NCERT Theorem 2 both composites are continuous.