Advanced Calculus : MATH 410
نویسنده
چکیده
It should be evident to you that (1.5) is equivalent to (1.3). Visually, if the graph of a function f at (a, f(a)) either has no unique tangent line or has a vertical tangent line then f is not differentiable at the point a. A function that is differentiable at every point in Dom(f) is said to be differentiable. A function that is differentiable at every point in a set S ⊂ Dom(f) is said to be differentiable over S. It is easy to see that if f is differentiable at the point a then it is continuous at a. Indeed, for every x ∈ Dom(f) such that x 6= a one has the identity
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Advanced Calculus: MATH 410 Functions and Regularity
5.1. Functions. We now turn our attention to the study of real-valued functions that are defined over arbitrary nonempty subsets of R. The subset of R over which such a function f is defined is called the domain of f , and is denoted Dom(f). We will write f : Dom(f) → R to indicate that f maps Dom(f) into R. For every x ∈ Dom(f) the function f associates the value f(x). The range of f is define...
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