If both f, g : R R are discontinuous at 0, then the product function fg : R R isa)Discontinuous at 0b)Continuous at 0c)Continuous at R\{0}d)NoneCorrect answer is option 'B'. Can you explain this answer?

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If both f, g : R → R are discontinuous at 0, then the product function fg : R → R is

a)
Discontinuous at 0
b)
Continuous at 0
c)
Continuous at R\{0}
d)
None

Correct answer is option 'B'. Can you explain this answer?

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If both f, g : R →R are discontinuous at 0, then the product fun...

f Suppose f: R→ R and g: R→ R. Then the composition g f is defined by g f (x) = g(f (x)).

We assume that f is continuous at p and that g is continuous at f (p). So suppose that (xi)→ p. Then (f (xi))→ f (p) and then (g(f (xi)))→ g(f (p)) which is what we need.

Examples

Clearly the identity function which x ↦ x is continuous.

Hence, using the above, any polynomial function is continuous and hence any rational function (a ratio of polynomial functions) is continuous at any point where the denominator is non-zero.

We will see later that functions like √, sin, cos, exp, log, ... are continuous. It follows that , for example sin2(x + 5), exp(-x2), √(1 + x4), ... are continuous since they are made by composing continuous functions.

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If both f, g : R →R are discontinuous at 0, then the product function fg : R →R isa)Discontinuous at 0b)Continuous at 0c)Continuous at R\{0}d)NoneCorrect answer is option 'B'. Can you explain this answer?

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