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Show that the Function Defined by F(X) = |Cos X| is a Continuous Function. - Mathematics

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

 
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Solution

The given function is f(x) = |cos x|

This function f is defined for every real number and f can be written as the composition of two functions as,

f = g o h, where g(x) = |x| and h(x) = cos x

Therefore, g is continuous at x = 0

From the above three observations, it can be concluded that g is continuous at all points.

(x) = cos x

It is evident that h (x) = cos x is defined for every real number.

Let be a real number. Put x = c + h

If x → c, then h → 0

(c) = cos c

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

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

Therefore, f(x) =(goh)(x) = g(h(x)) = g(cos x) = |cos x| is a continuous function.

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APPEARS IN

NCERT Class 12 Maths
Chapter 5 Continuity and Differentiability
Q 32 | Page 161
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