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Continuity of Composite Functions

If f(x) is co...

Question

If f(x) is continuous and g(x) is discontinuous at x = a, then the product function h(x) = f(x)g(x) must be discontinuous at x = a.

A

False

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B

True

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Q. If f(x) is continuous and g(x) is discontinuous at x = a, then the product function h(x) = f(x)g(x) must be discontinuous at x = a.

f (x) = sin x + [x]

is discontinuous at

x = 0

because Reason: If

g (x)

is continuous and

h (x)

is discontinuous at

x = a

g (x) + h (x)

will necessarily be discontinuous at

x = a

f (x)

be a continuous function and

g (x)

be a discontinuous function, then

f (x) + g (x)

is discontinuous function.

Q. Consider the functions

f (x) = s i n (x - 1)

g (x) = {cot}^{- 1} [x - 1]

Assertion: The function

F (x) = f (x) . g (x)

is discontinuous at

x = 1

f (x)

is discontinuous at

x = a

g (x)

is also discontinuous at

x = a

then the product function

f (x) . g (x)

is discontinuous at

x = a

Q. Statement $1 :$

f (x) = sin x + [x]

is discontinuous at

x = 0,

[.]

denotes the greatest integer function.
Statement $2 :$ If

g (x)

is continuous and

h (x)

is discontinuous at

x = a,

g (x) + h (x)

will necessarily be discontinuous at

x = a .

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