Question

Discuss the continuity of the function f given by
$straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell space space straight x comma space if space straight x greater or equal than 0 end cell row cell straight x squared comma space if space straight x less than 0 end cell end table close$

Solution

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

Function f is defined for all real numbers.
Now domain of f is divided into three disjoint subsets
D₁ = {.x ∈ R : x < 0}, D₂ = { 0}, D₃ = [x ∈ R : x > 0} of the real line.
Now three cases arise :
Case I : Let c ∈ D1 In this case f(x)=x²

therefore space Lt with straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals space Lt with straight x rightwards arrow straight c below space straight x squared equals straight c squared Also space space space space space straight f left parenthesis straight c right parenthesis equals straight c squared 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

Case II : Let c ∈ D₃. In this case f(x)=x

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

∴ f is continuous at x = c.
But c is any point of D₃.
∴ f is continuous in D₃.
Case III : We discuss continuity of f at x=0where f(x)=x²

Lt with straight x rightwards arrow 0 below straight f left parenthesis straight x right parenthesis equals Lt with straight x rightwards arrow 0 below straight x squared equals left parenthesis 0 right parenthesis squared equals 0

Now f is defined at x = 0
and f(0) = 0

therefore space Lt with straight x rightwards arrow 0 below straight f left parenthesis straight x right parenthesis equals straight f left parenthesis 0 right parenthesis equals 0

∴ f is continuous at x = 0.
From three cases, it is clear that f is continuous at every point of domain and so f is continuous function.

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