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Let f(x+y) = f(x) + f(y) for all x and ...

Let ` f(x+y) = f(x) + f(y)` for all x and y. If the function f (x) is continuous at x = 0 , then show that f (x) is continuous at all x.

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To show that the function \( f(x) \) is continuous at all \( x \) given that \( f(x+y) = f(x) + f(y) \) for all \( x \) and \( y \), and that \( f(x) \) is continuous at \( x = 0 \), we can follow these steps: ### Step 1: Understand the Functional Equation The functional equation \( f(x+y) = f(x) + f(y) \) suggests that \( f \) is a linear function. This is a property of Cauchy's functional equation. ### Step 2: Evaluate \( f(0) \) Let’s set \( x = 0 \) and \( y = 0 \) in the functional equation: \[ ...
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