Question

Discuss the continuity of the function f, where f is defined by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell 2 straight x comma end cell row cell 0 comma end cell row cell 4 straight x comma end cell end table close table row cell if space straight x less than 0 space space space space space space end cell row cell if space 0 less or equal than straight x less or equal than 1 end cell row cell if space straight x greater than 1 space space space space space space end cell end table$

Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell 2 straight x comma end cell row cell 0 comma end cell row cell 4 straight x comma end cell end table close table row cell if space straight x less than 0 space space space space space space space end cell row cell if space 0 less or equal than straight x less or equal than 1 end cell row cell if space straight x greater than 1 space space space space space space space end cell end table

Function f is defined at all points of the real line.
When x < 0, we have f(x) = 2 x, which being a linear polynomial. is continuous.

At space straight x equals 0 space space space space space space space space Lt with straight x rightwards arrow 0 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of minus below left parenthesis 2 straight x right parenthesis equals 0 space space space space space space space space Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of plus below left parenthesis 0 right parenthesis equals 0 Also space space space space space space space space space straight f left parenthesis 0 right parenthesis equals 0 therefore space space space Lt with straight x rightwards arrow 0 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 to the power of plus below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 0 right parenthesis

∴ f is discontinuous at x = 1
When x > 1, we have f(x) = 4 x, which being a linear polynomial is continuous.

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