Discuss the continuity of the cosine function.

Let f(x) = cos x

Formula used:

f(x) is continuous at x = c where c is any real number

if L.H.L = R.H.L = f(c)

Take L.H.L

Put x = c – h

x → c ⇒ c – h → c ⇒ h → c – c ⇒ h → 0

{∵ cos (A – B) = cos A cos B + sin A sin B}

= cos c (1) + (0) sin c

= cos c………………(1)

Take R.H.L

Put x = c + h

x → c ⇒ c + h → c ⇒ h → c – c ⇒ h → 0

{∵ cos (A + B) = cos A cos B – sin A sin B}

= cos c (1) – (0) sin c

= cos c………………(2)

f(x) = cos x

⇒ f(c) = cos c………………(3)

From (1), (2) and (3):

L.H.L = R.H.L = f(c)

⇒ f(x) is continuous for any real number

Hence, cos x is continuous