Show that f is continuous at every point in R

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A function f: R → R that satisfies the equation f(x + y) = f(x) + f(y) and is continuous at x = 0 can be shown to be continuous at every point in R. The key is to use the property of limits, specifically that if the limit of f(x) as x approaches x0 equals l, then the limit of f(x0 + h) as h approaches 0 also equals l. By applying this property and the functional equation, it can be demonstrated that f must be continuous at any point x0 in R. The continuity at x = 0 serves as a foundational point to extend continuity throughout the entire real line. Thus, the function f is continuous everywhere in R.
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Suppose a function f : R → R satisfy f(x + y) = f(x) + f(y) and f is continuous
at x = 0: Show that f is continuous at every point in R.

(Hint: Using the fact that
lim f(x) = l implies
x→x0
limf(x0+h)= l
h→0 )
 
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acacia89 said:
Suppose a function f : R → R satisfy f(x + y) = f(x) + f(y) and f is continuous
at x = 0: Show that f is continuous at every point in R.

(Hint: Using the fact that
lim f(x) = l implies
x→x0
limf(x0+h)= l
h→0 )

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