Question

Discuss the continuity of the function defined by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x plus 2 comma space space space space space if space straight x less than 0 end cell row cell negative straight x plus 2 comma space if space straight x greater than 0 end cell end table close$

Solution

Function f is defined for all real numbers except 0. Therefore domain of f is D₁ ∪ D₂ where
D₁ = {.x∈R : r < 0 }, D₂ = {x ∈ R : x > 0}
Now two cases arise :
Case 1: Let c ∈ D₁. In this case f(x) = x + 2.
$therefore space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below left parenthesis straight x plus 2 right parenthesis equals straight c plus 2 Also space space space space space space space straight f left parenthesis straight c right parenthesis equals straight c plus 2 therefore space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis$
∴ f (x) is continuous at x = c
But c is any point of D₁.
∴ f is continuous in D₁.
Case II : Let c ∈ D₂. In this case f(x) = –x + 2.
$therefore space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow straight c below left parenthesis negative straight x plus 2 right parenthesis equals negative straight c plus 2 Also space space space space space space space straight f left parenthesis straight c right parenthesis equals negative straight c plus 2 therefore space space space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis straight c right parenthesis$
∴ f(x) is continuous at x = c.
But c is any point of D₂.
∴ f is continuous in D₂.
Now f is continuous at all points in the domain of f. f
∴ f is continuous.

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