Question

Is the function defined by $straight f left parenthesis straight x right parenthesis equals open curly brackets table attributes columnalign left end attributes row cell straight x plus 5 comma space if space straight x less or equal than 1 end cell row cell straight x minus 5 comma space if space straight x greater than 1 end cell end table close$
a continuous function?

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 5 comma space if space straight x less or equal than 1 end cell row cell straight x minus 5 comma space if space straight x greater than 1 end cell end table close

Function f is defined at all points of the real line.
Let c be any real number.
Three cases arise :
Case I ; c < 1

space space space 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 straight f left parenthesis straight x plus 5 right parenthesis equals straight c plus 5 space space space space space space space space space space space space space straight f left parenthesis straight c right parenthesis equals straight c plus 5 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 II: c > 1

space space space 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 straight f left parenthesis straight x minus 5 right parenthesis equals straight c minus 5 space space space space space space space space space space space space space straight f left parenthesis straight c right parenthesis equals straight c minus 5 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: c = 1

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 straight f left parenthesis straight x plus 5 right parenthesis equals 1 plus 5 equals 6 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 straight f left parenthesis straight x minus 5 right parenthesis equals 1 minus 5 equals negative 4 therefore 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.

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