Question

Examine the continuity of the following:

x + sin x

Answer 1

Let f(x) = sin x

f(x) is defined at all points of R.

Let x₀ be an arbitrary point in R.

`lim_(x -> x_0) f(x) =  lim_(x -> x_0) (x + sin x)`

= x_o + sin x₀   ........(1)

f(x_o) = x_o + sin x_o   ........(2)

From equations (1) and (2) we get

`lim_(x -> x_0) f(x) = f(x_0)`

∴ At all points of R, the limit of f(x) exists and is equal to the value of the function.

Thus, f(x) satisfies ail conditions for continuity.

Therefore, f(x) is continuous at all points of f(x).