Introduction to Continuity

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Introduction

The property of continuity exists around various aspects of nature. The water flow in the rivers is continuous. The flow of time in human life is continuous, as we are getting older continuously. Similarly, in mathematics, we have the notion of the continuity of a function. If a function have a property that their graphs can be traced with a pencil without lifting the pencil from the page, then the functions is called continuous.

Intuitively, a function is continuous at a particular point if there is no break in its graph at that point.
let’s consider some examples to realize the intuitive notion of what it means to be continuous at a point.

(a):\( f(x)=\frac{x^2-1}{x-1}\)(b):\( \small {f(x)= \begin{cases} x+1 & \text{for } x ≤ 2 \\ x+2 & \text{for } x > 2 \end{cases}} \)(c): \( \small { f(x)= \begin{cases} x+1 & \text{for } x \ne 2 \\ 4 & \text{for } x = 2 \end{cases}} \)
  1. In Figure (a). We see that the graph of f(x) has a hole at x=1. In fact, f(1) is undefined. So f(x) is NOT contnuous at x=1
  2. In Figure (b), f(2) is defined, but the function has a jump at x=2. So f(x) is NOT contnuous at x=2.
  3. In Figure (c), f(2) is defined, but the function has a gap at x=2. So f(x) is NOT contnuous at x=2.

Intuitive Definition of Continuity

Let 𝑓(𝑥) be a function defined at all values in an open interval containing a, defining "a" itself
Then we say that
function f(x) is continuous at x=a
if for a given number \(\epsilon > 0\), there exists a number \(\delta > 0\)such that
|f(x)-f(a)|<\(\epsilon\) whenever |x-a|<\(\delta\)

In this definition

  1. Given a number a, we choose δ-neighbourhood of a, δ is positive AND can be small enough as we like such that |x - a|< δ
  2. Now, we will try to find, ε-neighbourhood of f(a), ε is positive AND can be small enough as satisfied such that |f(x) - f(a)|< ε
  3. If a small change in δ implies a small change ε, then the limit exists at a.
  4. If a small change in δ implies a LARGE change ε, or vice versa, then the limit does NOT exist at a.

Example: Continuous at x=a

How to use the applet
  1. Given a number a, adjust δ-neighbourhood of a (drag the point a or the slider δ) , so that |x - a|< δ , where x is any point inside the δ-neighbourhood
  2. Try to find ε-neighbourhood of f(a) (largest distance from f(a) , such that |f(x) - f(a)|< ε , where f(x) is any point inside the ε-neighbourhood and the result is valid for all |x - a|< δ

    सबै |x - a|< δ को लागी |f(x) - f(a)|< ε  हुने गरि ε-zone बानउन सकिन्छ भने continuous exist हुन्छ ।

    Once a ε is found, any higher ε is always accepted.
    Once a δ is satisfied, any smaller δ is always accepted.
More Explanation
The intuitive definition says that
  1. determine a number δ>0
  2. take any x in the region, i.e. between a+δ and a−δ, then this x will be closer to a, that is |x-a|<δ
  3. identify the point on the graph that our choice of x gives, then this point on the graph will lie in the intersection of the ε region. This means that this function value f(x) will be closer to f(a) , that is |f(x)-f(a)|<ε
    Means
    if we take any value of x in the δ region then the graph for those values of x will lie in the ε region.
  4. Once a δ is found, any smaller delta is acceptable, so there are an infinite number of possible δ's that we can choose.
  5. the function is continuous at given x

Example: NOT Continuous at x=1

Empirical Definition of Limit

Let 𝑓(𝑥) be a function defined at all values in an open interval containing a, defining "a" itself
Then we say that function f(x) is continuous at x=a if
all of the following three conditions hold

  1. f(a) is defined
  2. \( \displaystyle \lim_{x\to a^{-}}f(x)\) exist = LHL
  3. \( \displaystyle \lim_{x\to a^{+}}f(x) \) exist = RHL
  4. LHL = RHL = f(a)
Examples

  1. A function \( f(x) = x^2 + 1 \) is continuous at 2, since
    \( \displaystyle \lim_{x \rightarrow 2} f(x) =5 =f(2)\)

  2. A function \( f(x) = \sqrt{4-x^2} \) is NOT continuous at 3
    since f(3) is not defined.

  3. A function \( f(x)=\frac{1}{x-2}\)
    is not continuous at 2 because f(2) is not defined.

NOTES
  1. The constant function, f (x) = c, \( \forall x \in R\) is continuous on R.
  2. The identity function, f (x) = x, \( \forall x \in R\) is continuous on R.
  3. The function f (x) = x n, \( x \in N\) is continuous on R.
  4. The function f (x) = sinx is continuous.

Example 1

Test the continuity of a function \( f(x)= \begin{cases} -x^2 +2 &\text { for } x \ne 2 \\ 1 &\text { for } x=2 \end{cases} \) at x=2.

Solution

The solution is as follows
At \( x=2^{-}\), the left hand limit is
\( \displaystyle \lim_{x \to 2^{-}} f(x)= \lim_{x \to 2^{-} } -x^2+2 =-2\) LHL
At \( x=2^{+}\), the right hand limit is
\( \displaystyle \lim_{x \to 2^{+}} f(x)= \lim_{x \to 2^{+}} = -x^2+2 =-2\) RHL
At x=2, the value of the function is
f(2) = 1 Functional Value
Since, limiting value and functional value are not equal, f(x) is not continuous at x=2.

Example 2

Test the continuity of a function \(f(x)=\frac{|x|}{x}\) for \(x\ne 0\) at x=0.

Solution

The solution is
At \( x=0^{-}\), the left hand limit is
\( \displaystyle \lim_{x \to 0^{-}} f(x)= \lim_{x \to 0^{-} } \frac{|x|}{x}= \lim_{x \to 0^{-} } \frac{-x}{x}=\lim_{x \to 0^{-} } -1=-1\) LHL
At \( x=0^{+}\), the right hand limit is
\( \displaystyle \lim_{x \to 0^{+}} f(x)= \lim_{x \to 0^{+} } \frac{|x|}{x}= \lim_{x \to 0^{+} } \frac{x}{x}=\lim_{x \to 0^{+} } 1=1\) RHL
At x=0, the value of the function is
\(f(0)= \frac{|x|}{x}=\frac{0}{0}=\) Indeterminate f(x)
At x = 0, the value of f(0) is not defined.
So, the function is NOT continuous at x=0.

Example 3

Is\( f(x)=\frac{x+4}{x-3}\) continuous at x =1? At x =3?

Solution

The solution is as follows
At x=1, the limit value is
\( \displaystyle \lim_{x \to 1} f(x)= \lim_{x \to 1}\frac{x+4}{x-3}=\lim_{x \to 1} \frac{1+4}{1-3} =\frac{5}{-2}\) This is Limit Value
At x=1, the value of the function is
\( \displaystyle \lim_{x \to 1} f(x)= \lim_{x \to 1} \frac{1+4}{1-3} =\frac{5}{-2}\) This is Functional Value
Since, limiting value and functional value are equal, f(x) is continuous at x=1.
Next,
At x=3, the limit value is
At \( x=3^{-}\), the left hand limit is
\( \displaystyle \lim_{x \to 3^{-}} f(x)= \lim_{x \to 3^{-}} \frac{x+4}{x-3}=\lim_{h \to 0 } \frac{(3-h)+4}{(3-h)-3}=\lim_{h \to 0 } \frac{7-h}{-h}=-\infty\) LHL
At \( x=3^{+}\), the right hand limit is
\( \displaystyle \lim_{x \to 3^{+}} f(x)= \lim_{x \to 3^{+}} \frac{x+4}{x-3}=\lim_{h \to 0 } \frac{(3+h)+4}{(3+h)-3}=\lim_{h \to 0 } \frac{7+h}{h}=\infty\) RHL
At x=3, the value of the function is
\( \displaystyle \lim_{x \to 3} f(x)= \lim_{x \to 3} \frac{x+4}{x-3}= \frac{3+4}{0}=\infty\) f(x)
Since, LHL ≠RHL, f(x) is NOT continuous at x=3.

Discontinuity of a function

When a function is not continuous at a point, then we can say it is discontinuous at that point. There are several types of behaviors that lead to discontinuities. These several types can be classified into two major types of discontinuity, they are given below.
  1. Removable discontinuity
    1. Missing point discontinuity
      Let 𝑓(𝑥) be a function defined at all values in an open interval containing a.
      Then we say that function f(x) has missing point discontinuity at x=a if
      1. f(a) is not defined
      2. LHL = RHL
      See Example (a)
    2. Isolated point discontinuity
      Let 𝑓(𝑥) be a function defined at all values in an open interval containing a.
      Then we say that function f(x) has issolated point discontinuity at x=a if
      1. f(a) is defined
      2. f(a)≠LHL = RHL
      See Example (c) and Ex-1
  2. Non-removable discontinuity
    1. Finite point discontinuity
      Let 𝑓(𝑥) be a function defined at all values in an open interval containing a.
      Then we say that function f(x) has finite point discontinuity at x=a if
      1. finite=LHL≠RHL=finite
        one-sided limits are both finite, yet not equal to each other
      See example (b), Ex-2
      One More Example is given below.
      Test the continuity of a function \( f(x)= \begin{cases} 4x-3 & x < 2\\ (x-3)^2 & x ≥ 2 \end{cases} \) at x=2.

      Solution 👉 Click Here

    2. Infinite discontinuity
      Let 𝑓(𝑥) be a function defined at all values in an open interval containing a.
      Then we say that function f(x) has infinite discontinuity at x=a if
      1. At least one of LHL or RHL is infinite
      See Ex-3
      One more example is given below
      At what point is the function \(f(x)=\frac{x+1}{(x-2)(x-3)}\) (i) discontinuous (ii) continuous ?

      Solution 👉 Click Here

    3. Oscillatory discontinuity
      Let 𝑓(𝑥) be a function defined at all values in an open interval containing a.
      Then we say that function f(x) has oscillatory discontinuity at x=a if
      the values of the function appear to be approaching two or more values simultaneously.
      1. consider a function \( f(x)=\sin \frac{1}{x} \)
      2. take \( x=\frac{1}{\frac{\pi}{2} (2n+1)};n \in Z \)
      3. f(x) jumps alternatively to 1 and -1 as x approaches to 0 because \( \frac{1}{x}=\frac{\pi}{2} (2n+1);n \in Z\)
      4. \( f(x)=\sin \frac{1}{x} \) has oscillating discontinuity at x=0

Find the type of discontinuity at x=2,3,6,7

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