In daily life we understand a continuous process as that takes place smoothly without interruption or abrupt change. Mathematically a function is understood to be continuous in an interval if its graph can be drawn on that interval without breaks. This means the graph can be drawn without removing the pencil from any point in the interval. When we plot experimental data and join the points with smooth curves, we assume the continuity of the relationship.

Let us study how continuity is defined in calculus and the techniques used in determining the continuity of functions.

Continuity Definition

A function f(x) is said to be continuous at x = a
if $\lim_{x\rightarrow a}f(x)$ = f(a)
This definition implies three conditions necessary for continuity.
1. Function is defined at x = a, that is f(a) exists.
2. $\lim_{x\rightarrow a}f(x)$ exists
3. $\lim_{x\rightarrow a}f(x)$ = f(a)

All the above three conditions are to be satisfied for a function to be continuous at x = a.

The partial continuities of a function f(x) at a point x = a are defined as follows:

A function f(x) is said to be continuous to the right of x = a if,
$\lim_{x\rightarrow a^{+}}f(x)$ = f(a)

and the function is said to be continuous to the left of x =a if,
$\lim_{x\rightarrow a^{-}}f(x)$ = f(a)

Continuity on an interval
A function is said to be continuous in an open interval (a, b), if it is continuous in every point x, such that a < x < b.

A function is said to be continuous in a closed interval [a, b], if the it continuous at every interior point and right continuous at x = a and left continuous at x = b.

Properties of Continuity

The Properties of continuity on combinations of functions are similar to the limit laws.

Suppose functions f and g are continuous at a, then the following function combinations are also continuous at x = a.
1. f + g                                                Sum of functions
2. f - g                                                 Difference of functions
3. cf    where c is a constant                 Scalar multiple of a function
4. fg                                                    Product of functions
5. $\frac{f}{g}$  if g(a) $\neq$ 0.                      Quotient of functions

The above property can be extended to continuity of functions on an interval. If functions f and g are continuous in an interval so are the function combinations f + g, f - g, cf, fg and $\frac{f}{g}$ (if g is never 0 in the interval).

Limits and Continuity

A limit symbol can be moved through a function symbol, if the function is continuous and the limit exists.

Suppose that $\lim_{x\rightarrow a}g(x)=b$ and f is continuous at b, then
$\lim_{x\rightarrow a}f(g(x))$ = $f(\lim_{x\rightarrow a}g(x))$ = f( b) = b
This property allows the order to two symbols to be reversed.

This leads to the continuity property of function composition.

Suppose g(x) is continuous at a, and f(x) is continuous at g(a) then the function composition f o g is continuous at x =a.

In other words, g is continuous at a and f is continuous at g(a) ⇒ (f o g)(x) is continuous at x = a.

Uniform Continuity

Uniform continuity is considered to be stronger than the general continuity. It is defined as follows:

A function f defined on a domain D is said to be uniformly continuous if and only if for every ∈ > 0, there exists a δ such that for every x, y in domain D it is true that

|x - y| < δ ⇒ |f(x) - f(y)| < ε.

In the case of ordinary continuity, it is possible to find a δ for a known value of ∈ by fixing the value of one of the variables x or y as a constant.

But for the function to be uniformly continuous it is necessary to find a δ for all values of x and y in the domain D or in a specified interval I.

Continuous Functions

The following functions are continuous at every point in their domain.
1. Polynomials P(x) = anxn + an-1xn-1 +------------ + a1x + a0.
2. Rational functions f(x) = $\frac{p(x)}{q(x)}$ q(x) ≠ 0
3. Radical functions h(x) = $\sqrt[n]{x}$
4. Trigonometric functions Sin x, Cos x, Tan x, Csc x, Sec x, Cot x.
5. Inverse Trigonometric functions Sin-1 x , Cos-1 x , Tan-1 x
6. Exponential and logarithmic functions f(x) = ex , g(x) = ax , h(x) = ln x

Continuous Functions Examples

Solved Examples

Question 1: Find the intervals in which the function f(x) = $\frac{sin(x)+ln(x)}{x^{2}-4}$ is continuous.
Solution:

f(x) can considered as the quotient of two functions g(x) and h(x) where g(x) = sin x + ln x and h(x) = x2 - 4.
g(x) is a sum of two functions sin x and ln x where sin x is continuous for all reals and ln x is continuous in its domain (0, ∞)
Thus using the sum property of continuity g(x) is continuous in (0, ∞).
h(x) is a quadratic polynomial which is continuous every where, except points make denominator zero.
h(x) = 0 when x2 = 4 or x = ± 2

Thus applying the quotient property of continuity f(x) is continuous in (0, 2) and (2, ∞).

Question 2: Determine the values of the constants a and b that make f(x) continuous.
f(x) = $\left\{\begin{matrix} \frac{4sinx}{x} & if & x<0\\ acosx & if & x=0\\ b & if & x>0 \end{matrix}\right.$
Solution:

We need to check the continuity of f(x) at x = 0 as the function is continuous elsewhere.
$\lim_{x\rightarrow 0^{-}}f(x)$ = $\lim_{x\rightarrow 0^{-}}$$\frac{4sinx}{x}$  = 4         Applying limit formula
$\lim_{x\rightarrow 0^{+}}f(x)$ = $\lim_{x\rightarrow 0^{+}}b$   = b
For the limit to exist, $\lim_{x\rightarrow 0^{-}}f(x)$ = $\lim_{x\rightarrow 0^{+}}f(x)$
Hence b = 4  and $\lim_{x\rightarrow 0}f(x)$ = 4
For the function to be continuous at x = 0, f(0) = $\lim_{x\rightarrow 0}f(x)$
f(0) = a cos(0) = a                   Function definition at x = 0
Hence a = 4.
Thus the values a = b = 4 make f(x) defined piecewise continuous.

Question 3: Use continuity to evaluate the limit: $\lim_{x\rightarrow 2}$ 2x2 - 5x + 8.
Solution:

The  function f(x) = 2x2 - 5x + 8  is a quadratic polynomial and hence continuous for all real values of x.
Hence $\lim_{x\rightarrow 2}$ 2x2 - 5x + 8 = f(2)

= 2(2)2 -5(2) + 8 = 8 -10 + 8 = 6.