Question

Examine the continuity of the following:

x . log x

Answer 1

Let f(x) = x log x

The function f(x) is defined in the open interval `(0, oo)` since log x is defined for x > 0.

Let x₀ be an arbitrary point in `(0, oo)`.

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

= x₀ log x₀

f(x₀) = x₀ log x₀

From equation (1) and (2) we have

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

∴ The limit of the function f(x) exists at x = x₀ and is equal to the value of the function.

Since x₀ is an arbitrary point the above is true for all points in `(0, oo)`.

∴ f(x) is continuous at all points of `(0, oo)`.