If f is continuous and g is a discontinuous function then f + g is discontinuous function.

Solution

if possible, let (f + g ) be continuous, and as f is continuous then
(f + g ) – f is a continuous function (∵ difference of two continuous function is continuous)
⇒ g is continuous, which is a contradiction
Hence (f + g ) is discontinuous.
Note : If f and g are discontinuous, then f + g ,fg need not be discontinuous.
For example
$left parenthesis straight i right parenthesis space Let space straight f left parenthesis straight x right parenthesis equals tan space straight x comma space straight g space left parenthesis straight x right parenthesis space equals space cos space straight x comma space are space discontinuous.$
But (fg) (x) = (tan x) (cos x) = sin x, is continuous
(ii) $straight f left parenthesis straight x right parenthesis equals 1 over straight x equals tan space straight x comma space straight g left parenthesis straight x right parenthesis equals cos space straight x comma space are space discontinous.$
but (f + g) (x) = 0, is continuous at x = 0

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Continuity And Differentiability

If f is continuous and g is a discontinuous function then f + g is discontinuous function.

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