Continuity and Differentiability


The property of continuity is exhibited by various aspects of nature. The water flow in the rivers is continuous. The flow of time in human life is continuous i.e. you are getting older continuously. And so on. Similarly, in mathematics, we have the notion of the continuity of a function.

What it simply means is that a function is said to be continuous if you can sketch its curve on a graph without lifting your pen even once (provided that you can draw good). It is a very straightforward and close to accurate definition actually. But for the sake of higher mathematics, we must define it in a more precise way. That’s what we are going to do in this section. So let’s jump into it!

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Definition of Continuity

A function f(x) is said to be continuous at a point x = a, in its domain if the following three conditions are satisfied:

  1. f(a) exists (i.e. the value of f(a) is finite)
  2. Limx→a f(x) exists (i.e. the right-hand limit = left-hand limit, and both are finite)
  3. Limx→a f(x) = f(a)

The function f(x) is said to be continuous in the interval I = [x1,x2] if the three conditions mentioned above are satisfied for every point in the interval I.

However, note that at the end-points of the interval I, we need not consider both the right-hand and the left-hand limits for the calculation of Limx→a f(x). For a = x1, only the right-hand limit need be considered, and for a = x2, only the left-hand limit needs to be considered.

Some Typical Continuous Functions

  • Trigonometric Functions in certain periodic intervals (sin x, cos x, tan x etc.)
  • Polynomial Functions (x2 +x +1, x4 + 2….etc.)
  • Exponential Functions (e2x, 5ex etc.)
  • Logarithmic Functions in their domain (log10x, ln x2 etc.)


If any one of the three conditions for a function to be continuous fails; then the function is said to be discontinuous at that point. On the basis of the failure of which specific condition leads to discontinuity, we can define different types of discontinuities.

Jump Discontinuity

In this type of discontinuity, the right-hand limit and the left-hand limit for the function at x = a exists; but the two are not equal to each other. It can be shown as: $$ { Lim_{x\rightarrow{a^+}}  f(x) \neq  Lim_{x\rightarrow{a^-}}  f(x) } $$


Infinite Discontinuity

The function diverges at x = a to give it a discontinuous nature here. That is to say, $$ {\text{f(a) is not defined}} $$ Since the value of the function at x = a tends to infinity or doesn’t approach a particular finite value, the limits of the function as x → a are also not defined.


Point Discontinuity

This is a category of discontinuity in which the function has a well defined two-sided limit at x = a, but either f(a) is not defined or f(a) is not equal to its limit. The discrepancy can be shown as: $$ { Lim_{x\rightarrow{a}}  f(x) \neq f(a) } $$ This type of discontinuity is also known as a Removable Discontinuity since it can be easily eliminated by redefining the function in such a way that, $$ { f(a) = Lim_{x\rightarrow{a}}  f(x) } $$


Solved Examples for You

Question: Let a function be defined as f(x) =

5 – 2x for x < 1
3 for x = 1
x + 2 for x > 1

Is this function continuous for all x?

Solution: Since for x < 1 and x > 1, the function f(x) is defined by straight lines (that can be drawn continuously on a graph), the function will be continuous for all x ≠ 1. Now for x = 1, let us check all the three conditions:

–> f(1) = 3 (given)

–> Left-Hand Limit:
\( {= Lim_{x\rightarrow{1^-}}  f(x)} \)
\( {= Lim_{x\rightarrow{1^-}} (5 – 2x)} \)
\( {= 5 – 2\times{1}} \)
\( {= 3} \)

–> Right-Hand Limit:
\( {= Lim_{x\rightarrow{1^+}}  f(x)} \)
\( {= Lim_{x\rightarrow{1^+}} (x + 2)} \)
\( {= 1 + 2} \)
\( {= 3} \)

–> \( {Lim_{x\rightarrow{1^-}}  f(x) = Lim_{x\rightarrow{1^+}}  f(x) = 3 = f(1)} \)

Thus all the three conditions are satisfied and the function f(x) is found out to be continuous at x = 1. Therefore, f(x) is continuous for all x.

This concludes our discussion on the topic of continuity of functions. Continuous functions are very important as they are necessarily differentiable at every point on which they are continuous, and hence very simple to work upon.

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