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Question
Examine the continuity of the following:
x . log x
Solution
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)`.
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