A function is said to be continuous at a point a if the following statements hold:
- the function f is defined at a
- the limit limx→a f(x) exists
- the limit is equal to f(a), limx→a f(x)=f(a)
Continuity of a function can also be expressed more compactly by the statement: f(x)→f(a) as f→a
Some examples of functions with discontinuity at particular points include:
We can plot these functions to get a better understanding of where the discontinuity lay.
import numpy as np
import matplotlib.pyplot as plt
def f1(x):
return 1.0 / x ** 2.0
def f2(x):
return (x + 4.0) / ((x - 1.0) * (x - 8.0))
def f3(x):
return 1 / (x - 1)
fx1 = np.linspace(-2, 2, 1000)
fx2 = np.linspace(6, 10, 100)
fx3 = np.linspace(0, 2, 100)
plt.figure(figsize=(14,5))
plt.subplot(131)
plt.plot(fx1, f1(x1), fx1, f1(fx1))
plt.xlim([-1, 1])
plt.ylim([-500, 9500])
plt.subplot(132)
plt.plot(fx2, f2(fx2), fx2, f2(fx2))
plt.subplot(133)
plt.plot(fx3, f3(fx3), fx3, f3(fx3))
plt.show()
Although matplotlib
plots the line without any breaks, we see each graph has a point where the lines blow up and quickly approaches infinity in either the positive or negative direction.
Interestingly, and somewhat confusingly, although the functions above have a point of discontinuity, they are still said to be continuous functions. The key to this difference is that a function must be defined for all values of a. Because the functions cannot be defined at particular values of x due to dividing by 0, the functions are still continuous.
All polynomials are continuous anywhere in range (−∞,∞). Also, any rational function, which has the form P(x)Q(x) is continuous everywhere except for when Q(x)=0.
Knowing the continuity of a function helps evaluate limits much quicker. For example, consider the function:
The function is rational, thus we know it is continuous everywhere except when Q(x)=0. Thus, x≠−5 and we can solve the limit directly:
Intermediate Value Theorem¶
The Intermediate Value Theorem states that if a function f is continuous in a closed interval [a,b] and n is a number between f(a) and f(b) (where f(a)≠f(b)), then there exists a number c in the interval (a,b) such that f(c)=n.
The Intermediate Value Theorem can be applied when locating the root of an equation. For example, we can locate the root of the following equation within the interval [1,2] (should one exist).
Plotting the function first can help us determine if there is a root in the given interval.
def f4(x):
return x ** 4 + x - 3
xvals = np.linspace(1, 2, 100)
y1 = (0, 0)
plt.plot(xvals, f4(xvals))
plt.plot((0, 1.15), (0, 0), 'r--')
plt.plot((1.15, 1.15), (0,-5), 'r--')
plt.xlim([1, 2])
plt.ylim([-5, 15])
plt.show()
It appears the equation does have a root located in the interval [1,2]. We can also show this with the Intermediate Value Theorem, we take a=1, b=2 and n=0:
Therefore, f(1)<0<f(2) and n is a number between f(1) and f(2). We can get a more accurate interval of where the root is located by setting a tighter interval around the location of the root and applying the Intermediate Value Theorem. Judging by the graph, the root appears to be located in the interval [1.1,1.2], which we can verify by setting a=1.1, b=1.2 and n=0.
(f4(1.1), f4(1.2))
Thus, we now know the root is located in the interval [−0.44,0.27].
References¶
Continuous function. (2017, October 11). In Wikipedia, The Free Encyclopedia. From https://en.wikipedia.org/wiki/Continuous_function
Strang, G. (2010). Calculus. Wellesley, MA: Wellesley-Cambridge.