Question

Discuss the continuity of the function f defined by

$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x plus 2 comma space if space straight x less or equal than 1 end cell row cell straight x minus 2 comma space if space straight x greater than 1 end cell end table close$

Solution

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

The function f is defined at all points of the real line.
Let c be any real number
Case I: lf c < 1, then f(c) = c + 2.

therefore space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x plus 2 right parenthesis equals straight c plus 2 therefore space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis

∴ f is continuous at all real numbers less than 1.

Case II: If c > 1, then f(c) = c – 2.
$therefore space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below left parenthesis straight x minus 2 right parenthesis equals straight c minus 2 equals straight f left parenthesis straight c right parenthesis therefore space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis$
∴f is continuous at all points x > 1.
Case III : If c = 1, then
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∴ f is not continuous at x = 1
∴ x = 1 is the only point of discontinuity of f.
∴ f is not a continuous function.

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