My English lesson

My English lesson Derivability and continuity of a function DEFINITION BE CAREFUL! B It is i indifferent to speak of «derivative at x0 or «derivative at the point of abscissa x0 . We will be using both expressions in the following pages. We also often find in the literature the less rigorous, but immediate, expression «derivative at the point x0 . A function y = f(x), defined at least in the interval [a ; b], is derivable in a point x0 [a ; b], if there exists and is finite the limit: f( x0 + h) f(x0) f (x0) = lim _______________ h h 0 The real number f (x 0) is called the derivative of the function f at the point of abscissa x0. We saw in the previous unit that the derivative of a function at a point represents the direction of the tangent-line to the graph at that point: it is, if it exists, its angular coefficient. Then if the function f is derivable at all points of an interval (or at a ray or at the entire real axis), we can define the derivative function y = f (x), which associates with each number x of the interval the derivative of f(x) at the corresponding point. Let us now analyze what relationships exist between the derivability and continuity of a function at a given point and, in general, in an interval. We can show that if a function is derivable at a point of abscissa x0, then at that point it is also continuous. In Unit 2, we stated a theorem (continuity at a point) according to which for a function f to be continuous at the point of abscissa x0 it is necessary and sufficient that: lim f(x 0 + h) f(x 0) = 0 h 0 h Since we know that the function is derivable in x0, we can multiply by __ (which h equals 1) and apply the limit theorem of a product. We thus obtain: f(x 0 + h) f(x 0) lim f(x 0 + h) f(x 0) = lim _____________ h = f (x 0) lim h = f (x 0) 0 = 0 h h 0 h 0 h 0 This means that the function y = f(x) is continuous for x = x0. However, the opposite is not true: there are continuous functions at a point, but non-derivable there. For example, let us consider the function y = |x|. This is always defined and continuous, and its graph is formed by two rays of extremes the origin: for x 0 the ray has angular coefficient +1. y O x The limit of the incremental ratio of the function y = |x| in x0 = 0 is: 1 | h| lim _ = +1 h 0 y 1 O 276 1 x h if h 0 (right limit) it doesn t exist if h = 0 The function y = |x| has no derivative at x = 0 because, as x tends to 0, the left and right limits of the incremental ratio are different. The graph of its derivative for x 0 is in the figure opposite. At x = 0 the derivative function is not defined. The point x0 = 0 is called the angular point.

Il Maraschini-Palma - volume 5
Il Maraschini-Palma - volume 5