My English lesson Thus, the function is continuous in R if m = 7, and then: NOW IT S YOUR TURN N x2 x LI x a), we have that f (x) Jl. 1 1 O 1 x 1 1 1O 1 x 1 A b. a. DEFINITION A function is said to be continuous in an interval (or in its entire set of definition) if it is continuous at every point. It is important to know the intervals in which a function is continuous because at the points of these intervals it is not necessary to calculate the limit (being equal to the value of the function at that the point): the calculation of the limit is only performed at the points of discontinuity or when the independent variable x tends to infinity. example O Every constant function y = k is continuous in R (fig. a.). In fact, whatever is a R: lim k = k x a O The identical function y = x is continuous in R (fig. b.). In fact, for any real number a: lim x = a x a y k O a. y=k a x y=x y a O a x b. Some operations with continuous functions result in another continuous function. THEOREM (operations with continuous functions) If y = f(x) and y = g(x) are two continuous functions at the point a, then: Q the sum function f(x) + g(x) is continuous at the point a; Q the product function f(x) g(x) is continuous at the point a; Q the opposite function f(x) (symmetrical with respect to the x-axis) is continuous at the point a; f(x) Q the quotient function ____ (if a is not a zero for the function g) is cong(x) tinuous at the point a; Q the absolute value function |f(x)| is continuous at the point a. Consequently: any polynomial function y = anxn + ... + a2x2 + a1x + a0 is continuous in R. p(x) Q every fractional rational fraction y = _ is continuous in R {x q(x) = 0}. q(x) Q 171