Discuss the continuity of the following functions :

f(x) = sin x cos x

Idea : If f and g are two functions whose domains are same and both f and g are everywhere continuous then :

i) f + g is also everywhere continuous

ii) f – g is also everywhere continuous

iii) f*g is also everywhere continuous

iv) f/g is also everywhere continuous for all R except point at which g(x) = 0

∵ f(x) = sin x × cos x

It is a purely trigonometric function

As sin x is continuous everywhere and cos x is also continuous everywhere for all real values of x

As f(x) is nothing but product of two everywhere continuous function

∴ f(x) is also everywhere continuous.

We can see this through its graph which shows no point of discontinuity.

Fig : plot of sin x × cos x

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