Introduction to continuity
Definition and introductive examples
Continuity
Let f\left(x\right) be a function and a a real number in the domain of f\left(x\right). f\left(x\right) is continuous at x=a if:
\lim\limits_{x\to a}f\left(x\right)=f\left(a\right)
Consider the following piecewise function:
f\left(x\right)=\begin{cases} x^2 & x\not =2 \cr \cr 4 & x= 2 \end{cases}
We have:
- \lim\limits_{x\to 2}f\left(x\right)=\lim\limits_{x\to 2}x^2=4
- f\left(2\right)=4
Therefore:
\lim\limits_{x\to 2}f\left(x\right)=f\left(2\right)
f\left(x\right) is continuous at x=2.
Consider the following piecewise function:
f\left(x\right)=\begin{cases} x^2 & x\not =2 \cr \cr 5 & x= 2 \end{cases}
We have:
- \lim\limits_{x\to 2}f\left(x\right)=\lim\limits_{x\to 2}x^2=4
- f\left(2\right)=5
Therefore:
\lim\limits_{x\to 2}f\left(x\right)\neq f\left(2\right)
f\left(x\right) is not continuous at x=2.
We say that f\left(x\right) is continuous if f\left(x\right) is continuous at every value in its domain.
Continuity and graphs
If f\left(x\right) is continuous at x=a, then the graph of f\left(x\right) can be drawn around x=a without having to lift the pencil from the paper.
The following graph is the graph of a continuous function because the graph can be drawn by hand without having to lift the pencil from the paper.
The following graph is the graph of a function which is not continuous at one point. If one was to draw the graph by hand, then you would need to lift the pencil from the paper in order to completely draw the graph.
One-sided continuity
One-sided continuity
Let f\left(x\right) be a function. Then f\left(x\right) is left continuous at x=a if:
\lim\limits_{x\to a^-}f\left(x\right)=f\left(a\right)
Similarly, the function f\left(x\right) is right continuous at x=a if:
\lim\limits_{x\to a^+}f\left(x\right)=f\left(a\right)
Consider the greatest integer function:
f\left(x\right)=\lfloor x \rfloor
Let a be an integer. Then we have the following:
- \lim\limits_{x\to a^-}\lfloor x \rfloor=a-1
- \lim\limits_{x\to a^+}\lfloor x \rfloor =a
- \lfloor a \rfloor =a
Therefore the greatest integer function is not continuous at integer values. However, the greatest integer function is right-continuous at integer values.
Right-hand continuity of f\left(x\right)=\lfloor x\rfloor can be seen from its graph. Drawing the graph from right to left, one would not need to lift their pencil when approaching an integer x -value.
If f\left(x\right) is continuous at x=a, then f\left(x\right) is right continuous and left continuous at x=a.
Properties of continuous functions
Continuity of usual functions
Continuity of usual functions
Let f\left(x\right) be one of the following types of functions:
- Constant function
- Linear function
- Polynomial function
- Rational function
- Exponential function
- Logarithmic function
- Trigonometric function
- Inverse trigonometric function
Then f\left(x\right) is continuous at every point in its domain.
Consider the following function:
f\left(x\right)=\cos\left(x\right)
f\left(x\right) is a trigonometric function and its domain is all real numbers. Therefore, f\left(x\right) is continuous at all real numbers.
Operations with continuous functions
Continuity of addition of functions
Suppose f\left(x\right) and g\left(x\right) are continuous at x=a. Then the function \left(f+g\right)\left(x\right) is continuous at x=a.
The following two functions are continuous at all real numbers:
- f\left(x\right)=\sin\left(x\right)
- g\left(x\right)=x^3-12x
Therefore, the following function is continuous at all real numbers:
h\left(x\right)=\sin\left(x\right)+x^3-12x
Continuity of subtraction of functions
Suppose f\left(x\right) and g\left(x\right) are continuous at x=a. Then the function \left(f-g\right)\left(x\right) is continuous at x=a.
The following two functions are continuous at all real numbers:
- f\left(x\right)=\sin\left(x\right)
- g\left(x\right)=x^3-12x
Therefore, the following function is continuous at all real numbers:
h\left(x\right)=\sin\left(x\right)-\left(x^3-12x\right)
Continuity of multiplication of functions
Suppose f\left(x\right) and g\left(x\right) are continuous at x=a. Then the function \left(fg\right)\left(x\right) is continuous at x=a.
The following two functions are continuous at all real numbers:
- f\left(x\right)=\sin\left(x\right)
- g\left(x\right)=x^3-12x
Therefore, the following function is continuous at all real numbers:
h\left(x\right)=\sin\left(x\right)\left(x^3-12x\right)=\sin\left(x\right)x^3-\sin\left(x\right)12x
In particular, if c is a real number and f\left(x\right) is continuous at x=a, then the function c.f\left(x\right) is continuous at x=a.
Continuity of division of functions
Suppose f\left(x\right) and g\left(x\right) are continuous at x=a and that g\left(a\right)\not =0. Then the function \left(\dfrac{f}{g}\right)\left(x\right) is continuous at x=a.
The following two functions are continuous at x=1 :
- f\left(x\right)=\sin\left(x\right)
- g\left(x\right)=x^3-12x
Moreover, g\left(1\right)=-11\not=0. Therefore, the following function is continuous at x=1 :
h\left(x\right)=\dfrac{\sin\left(x\right)}{x^3-12x}
Continuity of composition of functions
Suppose f\left(x\right) at x=a and that g\left(x\right) is continuous at x=f\left(a\right). Then the composition function \left(g\circ f\right)\left(x\right) is continuous at x=a
The following two functions are continuous at all real numbers:
- f\left(x\right)=\sin\left(x\right)
- g\left(x\right)=x^3-12x
Therefore, the following composition functions are continuous at all real numbers:
- \left(g\circ f\right)\left(x\right)=\sin^3\left(x\right)-12\sin\left(x\right)
- \left(f\circ g\right)\left(x\right)=\sin\left(x^3-12x\right)
Intermediate value theorem
The graph of a continuous function can be drawn without having to lift the pencil from the paper. Intuitively, this implies that if the graph of continuous function is below the x -axis for one x -value and then above the x -axis for a different x -value, then at some point the graph of the function passed through the x -axis. This phenomenon is captured by the intermediate value theorem.
Intermediate Value Theorem
Let f\left(x\right) be a function which is continuous over an interval \left[a,b\right] and suppose that u is a real number such that:
\min\left(f\left(a\right),f\left(b\right)\right)\leq u \leq \max\left(f\left(a\right),f\left(b\right)\right)
There exists some number c such that a\leq c\leq b and:
f\left(c\right)=u
The Intermediate Value Theorem can be used to estimate x -intercepts of continuous functions.
Consider the following continuous function:
f\left(x\right)=x^3-2x-6
Observe the following:
- f\left(2\right)=8-4-6=-2
- f\left(3\right)=27-6-6=15
Observe further that -2\leq 0\leq 15. Therefore, by the intermediate value theorem there exists some real number c such that
- 2\leq c\leq 3
- f\left(c\right)=c^3-2c-6=0
