give an example of a function that has:
a) only one point of discontinuity
b) exactly two ponts of discontinuity
c) an infinite number of discontinuity
give an example of a function that is:
a) continuous at every point
b) discontinous at every point
i am confused on how to give these examples
think of a function which is undefined at one point.Usually that's a rational function, where it would evaluate to 0/0. So, try
y = (x^2-1)/(x-1).
y = x+1 for all real x, except where x=1. You can make it continuous by defining y(1)=2, which fills in the hole.
Extend this idea for two factors in the denominator
For c), try something like tan(x) which is discontinuities at all odd multiples of pi/2.
Every polynomial is continuous everywhere.
Something ghastly like
y=0 for rational x
y=1 for irrational x
is discontinuous everywhere. No matter how close you get to any rational number, there is an irrational number closer yet.
I can help you with those examples. Let's start with the functions that have a specific number of discontinuities.
a) To give an example of a function with only one point of discontinuity, we can consider the function f(x) = 1/x. This function is continuous everywhere except at x = 0, where it has a vertical asymptote and is not defined.
b) For a function with exactly two points of discontinuity, we can take the function g(x) = |x|. This function is continuous everywhere except at x = 0, where it has a sharp corner or "point" of discontinuity, and at x = -1, where it has a jump discontinuity.
c) An example of a function with an infinite number of discontinuities is the Dirichlet function, denoted as D(x), which is defined as follows:
D(x) = 1 if x is rational
D(x) = 0 if x is irrational
This function has a discontinuity at every point on the real number line since every point can be either rational or irrational.
Now let's consider functions that have different properties for continuity.
a) An example of a function that is continuous at every point is the identity function f(x) = x. This function is a straight line with a slope of 1 and is continuous for all real numbers.
b) An example of a function that is discontinuous at every point is the Dirichlet function, which I mentioned earlier. It has a discontinuity at every point, regardless of whether it is rational or irrational.
I hope these examples help clarify the concepts of functions with different numbers of discontinuities and different properties of continuity.
Sure! I will explain how to give examples of functions with the desired properties.
a) To find a function with only one point of discontinuity, you can consider a rational function. One example is:
f(x) = (x^2 - 1) / (x - 1)
In this case, there is a point of discontinuity at x = 1, where the function is not defined due to division by zero.
b) To find a function with exactly two points of discontinuity, you can use a piecewise function. For example:
f(x) = { 1, if x < 0,
0, if 0 ≤ x ≤ 1,
-1, if x > 1 }
In this case, there are two points of discontinuity at x = 0 and x = 1, where the function jumps from one value to another abruptly.
c) To find a function with an infinite number of discontinuities, you can use a trigonometric function. One example is:
f(x) = sin(1 / x)
In this case, there are infinite points of discontinuity as x approaches zero. The function oscillates rapidly between -1 and 1 as x approaches zero, making it discontinuous at all values of x = 1/n (where n is an integer).
a) To find a function continuous at every point, consider a polynomial function. For example:
f(x) = 2x^3 - 3x^2 + 4x - 5
In this case, the function is defined and continuous at every point in its domain, which includes all real numbers.
b) To find a function discontinuous at every point, you can use a Dirichlet function. For example:
f(x) = { 1, if x is rational,
0, if x is irrational }
In this case, the function takes different values for rational and irrational numbers, leading to discontinuity at every single value of x.