Question

Discuss the continuity of the function f, where/is defined by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell 3 comma space space if space 0 less or equal than straight x less or equal than 1 space space end cell row cell 4 comma space space if space 1 less than straight x less than 3 end cell row cell 5 comma space if space 3 less or equal than straight x less or equal than 1 space end cell end table close$

Solution

Here space straight f left parenthesis straight x right parenthesis equals open curly brackets table row cell 3 comma space space if space 0 less or equal than straight x less or equal than 1 end cell row cell 4 comma space space if space 1 less than straight x less than 3 end cell row cell 5 comma space if space 3 less or equal than straight x less or equal than 10 end cell end table close

Function f is defined at all points of the domain.
In the interval 0 < x < 1, we have f(x) = 3, which is constant and so it is continuous.

At space straight x equals 1 space space space space space space space space Lt with straight x rightwards arrow 1 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 to the power of minus below left parenthesis negative 3 right parenthesis equals 3 space space space space space space space space Lt with straight x rightwards arrow 1 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 1 to the power of plus below left parenthesis 4 right parenthesis equals 4 therefore space space space space Lt with straight x rightwards arrow 1 to the power of minus below straight f left parenthesis straight x right parenthesis not equal to Lt with straight x rightwards arrow 1 to the power of plus below straight f left parenthesis straight x right parenthesis

∴ f is discontinuous at x = 1.
In the interval 1 < x < 3, we have f(x) = 4, which is constant and so it is continuous.

At space straight x equals 3 space space space space space space space space Lt with straight x rightwards arrow 3 to the power of minus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 3 to the power of minus below left parenthesis 4 right parenthesis equals 4 space space space space space space space space Lt with straight x rightwards arrow 3 to the power of plus below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 3 to the power of plus below left parenthesis 5 right parenthesis equals 5 therefore space space space space Lt with straight x rightwards arrow 3 to the power of minus below straight f left parenthesis straight x right parenthesis not equal to Lt with straight x rightwards arrow 3 to the power of plus below straight f left parenthesis straight x right parenthesis

∴ f is discontinuous at x = 3.
In the interval 3 < x < 10, we have f(x) = 5, which is constant and so it is continuous.

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