Question

Let a function ƒ : R→ R satisfy the equation ƒ(x + y) = ƒ(x) + ƒ(y) for all x, y. If the function ƒ(x) is continuous at x = 0, then –

• A. ƒ (x) is discontinuous for all x
• B. ƒ (x) is continuous for all positive real x
• C. ƒ (x) is continuous for all x
• D. None of these

Answer 1

Answer: (C)

ƒ (x) is continuous for all x

Answer 2

Since ƒ(x) is continuous at x = 0,

∴ $\lim _ { x \rightarrow 0 }$ ƒ(x) = ƒ(0).

Take any point x = a, then at x = a

$\lim _ { \mathrm { x } \rightarrow \mathrm { a } }$ ƒ(x) = $\lim _ { h \rightarrow 0 }$ ƒ(a + h)

= $\lim _ { h \rightarrow 0 }$ [ƒ(1) + ƒ(h)]

[Q ƒ(x + y) = ƒ(x) + ƒ(y)]

= ƒ(1) + $\lim _ { h \rightarrow 0 }$ ƒ(h) = ƒ(1) + ƒ(0)

= ƒ(a + 0) = ƒ(1)

∴ ƒ(x) is continuous at x = a. Since x = a is any arbitrary point, therefore ƒ(x) is continuous for all x.