Let f be a function satisfying f(x+y)=f(x)f(y)−4−f(y) for all x,y∈ R. and limx→0f(x)=4. Discuss the continuity of f(x)

Given f(x+y)=f(x)f(y)−4−f(y) for all x x,y∈R Solving f(x+h)=f(x)f(h)−4−f(h) ∴limh→0f(x+h) =limh→0{f(x)f(h)−4−f(h)} =limh→0f(x)f(h)−limh→04−f(h) =f(x)limh→0f(h)−limh→04−f(h) =4f(x)−4−4 =4f(x) Thus limh→0f(x+h)f(x), If f(x)≠0. But f(x)=0 doesnot satisfy the given condition. Therefore, f(x)≠0. If f(x)=0, then limh→0f(x+h)=f(x) hence, f(x) is discontinous at x.

Expert Answer to "Let f be a function satisfying f(x+y)=f(x)f(y)−4−f(y) for..."

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2021-11-07

Let f be a function satisfying $f (x + y) = f (x) f (y) - \sqrt{4 - f (y)}$ for all $x, y \in R$ . and $lim_{x \to 0} f (x) = 4$ . Discuss the continuity of f(x)

Answer & Explanation

krolaniaN

Skilled2021-11-08Added 86 answers

Given
$f (x + y) = f (x) f (y) - \sqrt{4 - f (y)}$ for all x $x, y \in R$
Solving
$f (x + h) = f (x) f (h) - \sqrt{4 - f (h)}$
$∴ lim_{h \to 0} f (x + h)$
$= lim_{h \to 0} {f (x) f (h) - \sqrt{4 - f (h)}}$
$= lim_{h \to 0} f (x) f (h) - lim_{h \to 0} \sqrt{4 - f (h)}$
$= f (x) lim_{h \to 0} f (h) - lim_{h \to 0} \sqrt{4 - f (h)}$
$= 4 f (x) - \sqrt{4 - 4}$
$= 4 f (x)$
Thus $lim_{h \to 0} f (x + h) f (x)$ , If $f (x) \neq 0$ . But f(x)=0 doesnot satisfy the given condition. Therefore, $f (x) \neq 0$ . If f(x)=0, then $lim_{h \to 0} f (x + h) = f (x)$ hence, f(x) is discontinous at x.

Jeffrey Jordon

Expert2022-06-26Added 2605 answers

Answer is given below (on video)

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