Questions with answers on the continuity of functions with emphasis on rational and piecewise functions. The continuity of a function and its derivative at a given point is discussed. Graphical meaning and interpretation of continuity are also included.
Example 1: For what values of x are each of the following functions discontinuous?
Solution to Example 1
a) For x = 0, the denominator of function f(x) is equal to 0 and f(x) is not defined and does not have a limit at x = 0. Therefore function f(x) is discontinuous at x = 0.
b) For x = 2 the denominator of function g(x) is equal to 0 and function g(x) not defined at x = 2 and it has no limit. Function g(x) is not continuous at x = 2.
c) The denominator of function h(x) can be factored as follows: x2 -1 = (x - 1)(x + 1). The denominator is equal to 0 for x = 1 and x = -1 values for which the function is undefined and has no limits. Function h is discontinuous at x = 1 and x = -1.
d) tan(x) is undefined for all values of x such that x = ?/2 + k ? , where k is any integer (k = 0, -1, 1, -2, 2,...) and is therefore discontinuous for these same values of x.
e) The denominator of function j(x) is equal to 0 for x such that cos(x) - 1 = 0 or x = k (2 ?), where k is any integer and therefore this function is undefined and therefore discontinuous for all these same values of x.
f) Function k(x) is defined as the ratio of two continuous functions (with denominator x2 + 5 never equal to 0), is defined for all real values of x and therefore has no point of discontinuity.
g) l(x) = (x + 4)/(x + 4) = 1 . Hence lim l(x) as x approaches -4 = 1 = l(-4) . Function l(x) is continuous for all real values of x and therefore has no point of discontinuity.
Example 2: Find b such that f(x) given below is continuous?
Solution to Example 2
For x > -1, f(x) = 2 x 2 + b is a polynomial function and therefore continuous.
For x < -1, f(x)= -x 3 is a polynomial function and therefore continuous.
For x = -1
f(-1) = 2(-1) 2 + b = 2 + b
let us consider the left and right hand limits
limit from left of -1
Example 3: Find a and b such that both g(x) given below and its first derivative are continuous?
Solution to Example 3
Continuity of function g
For x > 2, g(x) = a x 2 + b is a polynomial function and therefore continuous.
For x < 2, g(x) = -2 x + 2 is a polynomial function and therefore continuous.