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

Explanation

Let `f(x) = | cos x|`
at x = 0
`f(0) = |cos 0 | =1`
L.H.L. = `underset(x to 0^(-))(lim)f(x) underset (h to 0)(lim)f(0-h)`
= `underset(h to 0 )(lim) |cos(-h)|=underset(h to 0)(lim) |1|=1`
R.H.L. = `underset(x to 0^(+))(lim)f(x) underset (h to 0)(lim)f(0+h)`
= `underset(h to 0 )(lim) |cos h|=underset(h to 0)(lim) |1|=1`
`:'` L.H.L. = f(0) = R.H.L.
`:'` f(x) is contnuous at x =0
We know that `|x|` and cos x are continuous for x 0.
`:.` |cos x | is continuous for x = 0
Therefore, `f(x) = | cos x |` is always a continuous function. Hence proved