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Continuity And Differentiability

Question
CBSEENMA12034567
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Let f be a function such that f(x + y) =f(x) +f(y), x,y∈ R
Show that if f is continuous at x = 0, then it is continuous everywhere.

Solution

Here f(x + y) = f(x) + f(y) ∀ x, y∈  R    ..(1)
Let f be continuous at x = 0
therefore space space space space space space space space space space space space 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 space space space space space space space space space rightwards double arrow space Lt with straight h rightwards arrow 0 below straight f left parenthesis 0 plus straight h right parenthesis equals straight f left parenthesis 0 right parenthesis rightwards double arrow space Lt with straight h rightwards arrow 0 below left square bracket straight f left parenthesis 0 right parenthesis plus straight f left parenthesis straight h right parenthesis right square bracket equals straight f left parenthesis 0 right parenthesis space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space left square bracket therefore space of space left parenthesis 1 right parenthesis right square bracket rightwards double arrow space Lt with straight x rightwards arrow 0 below straight f left parenthesis 0 right parenthesis plus Lt with straight x rightwards arrow 0 below straight f left parenthesis straight h right parenthesis equals straight f left parenthesis 0 right parenthesis space space rightwards double arrow space straight f left parenthesis 0 right parenthesis plus Lt with straight h rightwards arrow 0 below straight f left parenthesis straight h right parenthesis equals straight f left parenthesis 0 right parenthesis rightwards double arrow space Lt with straight h rightwards arrow 0 below straight f left parenthesis straight h right parenthesis equals 0 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis Let space straight c space be space any space real space number therefore space Lt with space space straight x rightwards arrow straight c below straight f left parenthesis straight x right parenthesis equals Lt with straight h rightwards arrow 0 below straight f left parenthesis straight c plus straight h right parenthesis equals Lt with straight h rightwards arrow 0 below left square bracket straight f left parenthesis straight c right parenthesis plus straight f left parenthesis straight h right parenthesis right square bracket space space space space space space space space space space space space space space space space space space space space space equals Lt with straight h rightwards arrow 0 below straight f left parenthesis straight c right parenthesis plus Lt with straight h rightwards arrow 0 below straight f left parenthesis straight h right parenthesis equals straight f left parenthesis straight c right parenthesis plus 0 space space space space space space space space space space left square bracket space therefore space of left parenthesis 2 right parenthesis right square bracket therefore space 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 is continuous at x = c
But c is any real number
∴ f is continuous ∀ .x ∈ R.

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