Show that the function defined by f(x) = cos(x²) is a continuous function.

Solution:

The given function is f(x) = cos(x²).

This function f is defined for every real number

and f can be written as the composition of two functions as,

f = goh, where g(x) = cosx and h(x) = x²

[since (goh)(x) = g(h(x)) = g(x²) = cos(x²) = f(x)]

It has to be proved first that g(x) = cosx and h(x) = x² are continuous functions.

It is evident that g is defined for every real number.

Let c be a real number.

Let g(c) = cos c.

Put x = c + h

If x→c , then h→0

lim_x→c g(x) = lim_x→c cosx = lim_h→0 cos(c + h)

= lim_h→0 [cos c cos h − sin c sin h]

= lim_h→0 (cos c cos h) − lim_h→0 (sin c sin h)

= cos c cos 0 − sin c sin 0

= cos c (1) − sinc (0) = cos c

∴lim_x→c g(x) = g(c)

Therefore, g(x) = cos x is a continuous function.

Let h(x) = x²

It is evident that h is defined for every real number.

Let k be a real number, then h(k) = k²

lim _x→k h(x) = lim _x→k x² = k²

⇒ lim _x→k h(x) = h(k)

Therefore, h is a continuous function.

It is known that for real-valued functions g and h, such that (goh) is defined at c, if g is continuous at c and if f is continuous at g(c), then (fog) is continuous at c.

Therefore, f(x) = (goh) (x) = cos(x²) is a continuous function

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NCERT Solutions Class 12 Maths - Chapter 5 Exercise 5.1 Question 31

Show that the function defined by f(x) = cos(x²) is a continuous function.g(x) = cos x is a continuous function

Summary:

The function defined by f(x) = cos(x²) is a continuous function