Continuous means without any break or sudden jump or holes at any point. In Mathematics, continuous function also has similar kind of definition. If a small change in the input of a function brings a small change in the output of that function, then we say that the function is continuous. For the function to be continuous at any point x=a, the function must be defined at that point and limiting values of f(x) when x approaches a, is equal to f(a). In other words we say, If f(x) is a function such that $\lim_{x \rightarrow a}$ f(x) = f(a), then f(x) is continuous at “a”.

Weierstrass definition (epsilon-delta definition) of continuous functions: A function ƒ is said to be continuous at the point c, if for any number $\epsilon$ > 0, however small, there exists some number $\delta$ > 0 such that for all x in the domain of ƒ with |x-c| < $\delta$, the value of ƒ(x) satisfies |f(x) –f(c)| < $\epsilon$.Some real life examples for continuous functions are: Temperature at various times of the day, height growth as a function of time, cost of the cab ride as a function of distance travelled, population growth in a city as a function of time, etc.

If f(x) is not continuous at x = a, we say that f(x) is discontinuous at x = a.

1. Let f(x)and g(x) be continuous functions on a domain D,then f(x) ± g(x) is continuous on D.

2. If f(x)and g(x) are continuous functions on a domain D and c is any real number,
then cf(x) is continuous on D.

3. If f(x)and g(x)are continuous functions on a domain D,then,f(x) · g(x) is continuous on D.

4. If f(x)and g(x) are continuous functions on a domain D,then, $\frac{f(x)}{g(x)}$ is continuous on D wherever g(x) $\neq$ 0.

5. Suppose that f(x)and g(x) are continuous functions on a domain D, then f(g(x))and g(f(x))
are continuous on some domain.

6. Differentiation represents nothing but the ratio by which f(x)changes for small change in x. It can be written as f'(x) = $\lim_{x \rightarrow 0}$ $\frac{\Delta y}{\Delta x}$ = $\frac{dy}{dx}$.

Every differentiable function f:(a, b) $\rightarrow$ R is continuous at all points in its domain.

The converse need not hold true which means a continuous function need not be differentiable.

For example, f(x) = |x| is continuous everywhere but not differentiable at x = 0. Thus all continuous functions need not have derivatives.
Similarly, we can also see that every continuous function is integrable. But, the converse does not hold true. Thus all continuous function on a closed interval [a, b] have anti-derivatives.
7.Intermediate Value Theorem: Let f be a function which is continuous on the closed interval [a,b]. Let d be a real number between f(a) and f(b); then there exists c $\epsilon$ [a, b] such that f(c) = d.

Suppose that a point “a” is in the domain of the function f such that, for all x in the domain of f, f(x) < f(a) then f is said to have a maximum value at x = a. Suppose that a point “a” is in the domain of the function f such that, for all x in the domain of f, f(x) > f(a) then f is said to have a minimum value at x = a.

8. Extreme Value Theorem: Suppose that f is a function which is continuous on the closed interval [a, b]. Then there exist real numbers c and d in [a, b]such that, f has a maximum value at x = c and f has a minimum value at x = d.

9. Polynomials, Rational Functions, Exponential functions and Logarithmic functions, Root Functions and Trigonometric functions are basic continuous functions which are always continuous everywhere in their domains.
To Prove that a function f is continuous at a point “a”, the following steps should be followed.

(i) f(a) must exist.

(ii) $\lim_{x \rightarrow a}$ f(x) should exist which means the left hand limit $\lim_{x \rightarrow a}$ - f(x) = $\lim_{x \rightarrow a}$ + f(x) which is the right
hand limit of function f(x) at the point a.

(iii) $\lim_{x \rightarrow a}$ f(x) = f(a).
A function is said to be piecewise continuous on an interval if the interval can be broken into a finite number of subintervals on which the function is continuous on each open subinterval without its endpoints and has a finite limit at the endpoints of each subinterval.
 
A function f(x) is said to be a piece-wise continuous function on [a, b] $\subset$ $\Re$, if there exists finite number of points a = $x_0 < x_1 < x2 <...
Piecewise ContinuousPiecewise Continuous Function
 
Piecewise continuous functions might not have vertical asymptotes. Removable and step discontinuities are the only possible types of discontinuities for a piecewise continuous function.
Let I $\subset$ R.A function f: I $\rightarrow$ R is absolutely continuous on I if for every $\varepsilon$ > 0, $\exists$ $\delta$ > 0, such that whenever a finite sequence of pairwise disjoint sub-intervals ($x_i$, $y_i$) of I satisfies

$\sum_k$ | $y_i - x_i$ | < $\delta$ then $\sum_k$ | f($y_i$) - f($x_i$) | < $\varepsilon$.

Every absolutely continuous function is continuous but converse does not hold. For example f(x)= $x^2$.
A function f is uniformly continuous on a set I $\subseteq$ R if and only if $\forall$ $\varepsilon$ > 0 $\exists$ $\delta$ > 0, $\forall$ x, y $\epsilon$ I,

|x - y| < $\delta$ $\rightarrow$ | f(x) - f(y)|< $\forall$.

Every uniformly continuous function is continuous, but the converse does not hold. For example f(x)= $\frac{1}{x}$. This function is continuous in its domain but is not uniformly continuous in its whole domain.
A function is said to be non continuous function if it is discontinuous. If f(x) is not continuous at x = a, we say that f(x) is discontinuous at x = a. For example, the function f(x) = $\frac{1}{x}$ is discontinuous at x = 0. The graph of this function is shown below.
Non Continuous Function
A function defined over the set of real numbers to the real numbers can be represented by a graph in the Cartesian plane. The function is continuous if, the graph is a single unbroken curve without any breaks or sudden jumps or holes at any point.
           

Continuous Function Graphs

If f(x) is not continuous at a point, we say that f(x) is discontinuous at that point.
                            
Graphs of Continuous Functions
Graph of Continuous Function

In the figure(i) above, the graph has a break at x = -3, so the function is discontinuous at x = -3.

Similarly, in the figure (ii), the graph has a jump at x = -1, thus the function is discontinuous at x = -1.

Solved Examples

Question 1: Prove that the function f(x) = $\frac{x^2 - 3x + 2}{x - 2}$ for x $\neq$ 2 and f(x) = 1 for x = 2 is a continuous function at x = 2.
Solution:
 
To prove that a function is continuous at x = 2 we have to prove that $\lim_{x \rightarrow 2}$ f(x) = f(2).

$\lim_{x \rightarrow 2}$ f(x) = $\lim_{x \rightarrow 2}$$\frac{x^2 - 3x + 2}{x - 2}$

                        = $\lim_{x \rightarrow 2}$$\frac{(x - 2)(x-1)}{x - 2}$
               
                        = $\lim_{x \rightarrow 2}$(x-1) = 2 - 1 = 1 = f(2)
                       
Thus the function is continuous at x = 2.
 

Question 2: Find the value of c which makes the function f(x) = 2x + c for x $\leq$ 1 and  $x^2$ + 3 for x > 1 continuous.
Solution:
 
Given that the function f(x) is continuous, the left hand limit is equal to the right hand limit at x = 1. Thus,

$\lim_{x \rightarrow 1 -}$ 2x + c = $\lim_{x \rightarrow 1 +}$$x^2$ + 3

2(1) + c = 1 + 3

2 + c = 4

C = 4 - 2 

C = 2.