Solution of the 8th Homework
نویسنده
چکیده
φ f (x) = f (x)− f (x0) x−x0 , x , x0 f ′(x0), x = x0. Now, by the differentiability of f at x0, we have lim x→x0 x∈X\{x0 } φ f (x) = lim x→x0 x∈X\{x0 } f (x) − f (x0) x − x0 = f (x0) = φ f (x0). This proves that φ f is continuous at x0. Moreover, (1.1) holds for all x ∈ X by definition of φ f . ((1.1) is clearly true when x , x0. When x = x0, both sides of (1.1) are zero.) Therefore (ii) follows.
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