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Question

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

A
f(x)=0 continuous for all x
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B
f(x) is continuous for all positive real x
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C
f(x) is continuous for all x
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D
None of these
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Solution

## The correct option is C f(x) is continuous for all xSince f(x) is continuous at x=0,∴limx→0f(x)=f(0)Take any point x=a, then at x=alimx→af(x)=limh→0f(a+h)=limh→0[f(a)+f(h)]=f(a)+limh→0f(h)=f(a)+f(0)=f(a+0)=f(a)∴f(x) is continuous at x=a.Since x=a is any arbitrary point, therefore f(x) is continuous for all x.

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