Let fx-y-fx-fy for all x- y- R- If fx is continue at x-0 show that fx is continuous at all x

If f( x ) is continuous at x then f(x+Δ x)=f(x Δ x)=f(x) Or, consider x→ 0 If f( x ) is continuous at 0 then f(+Δ x)=f( Δ x)=f( ...

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Byju's Answer

Standard XII

Mathematics

Theorems for Continuity

Let fx+y=fx...

Question

Let $f (x + y) = f (x) + f (y)$ for all $x, y \in R$ . If $f (x)$ is continue at $x = 0$ show that $f (x)$ is continuous at all $x$

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Solution

If $f (x)$ is continuous at $x$ then $f (x + Δ x) = f (x - Δ x) = f (x)$
Or, consider $x \to 0$
If $f (x)$ is continuous at $0$ then $f (+ Δ x) = f (- Δ x) = f (0)$
$f (x + y) = f (x) + f (y)$
Let $y = 0$
$f (x + 0) = f (x) + f (0)$
$f (0) = 0$
$f (x \pm Δ) = f (x) + f (\pm Δ) = f (x) + f (0) = f (x)$

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Let f(x+y)=f(x)+f(y) for all x, y∈R. If f(x) is continue at x=0 show that f(x) is continuous at all x

Iff(x)is continuous at x thenf(x+Δx)=f(x−Δx)=f(x)Or, consider x→0Iff(x)is continuous at 0 thenf(+Δx)=f(−Δx)=f(0)f(x+y)=f(x)+f(y)Let y=0f(x+0)=f(x)+f(0)f(0)=0f(x±Δ)=f(x)+f(±Δ)=f(x)+f(0)=f(x)

Let $f (x + y) = f (x) + f (y)$ for all $x, y \in R$ . If $f (x)$ is continue at $x = 0$ show that $f (x)$ is continuous at all $x$