Continuity

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Continuity at a Point
The formal definition of continuity at a point “a” should have a following three condition. Otherwise, a function f(x) is continuous at point “a” if the following three conditions are satisfied.

1. f(a) is defined or exists (the function is defined a)
2. f(x) exists (i.e. finite)
3. f(x) = f (a)
If any function fails from above three conditions at a point “a” is said to be a discontinuous at point “a”.
f(x) = (x2+x)/(x2-x)
The above equation satisfy the conditions of 1 and 2 but fails at condition 3 for a =0. So it is a discontinuous function.
If it is continuous at every point in the interval, a function is continuous on an interval.

Continuity
Continuity can be defined as a function f(x) is continuous at an every point “a” set. Finally if it is continuous at every point of its domain, f(x) is continuous. Let “a” be a point in the domain of the function f(x).
f(x) = f(a) (f is continuous at x =a)
For all real number “a” and this property is known as continuity.
Continuity can be simply said ...
... that the function is continuous on an interval. The graph should not have any jumps, gaps, or holes, if the function is continuous.
Continuity Functions

Continuity functions are the functions, the small changes in the input gives the small changes in the out put. The function which is not continuous is said to be a discontinuous. Topology is the one of the core concept of continuity of function.
Example for continuity function
f(t) is the function of growing tree at time t. so this function is continuous.

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Example for discontinuity function
m(t) is the function of the amount of money in a bank account at time. Where money is deposited or withdraw, the function jumps, so the function id discontinuity.
Continuity Equations
Continuity equation is the mathematical equation or statement that describes conservation of mass. For example the mass preserving in a continuum material body is,
M = L3 T-2

Continuity equation can be expressed in the form of integral or flex integral and it underlie more equations of transport such as convection diffusion equation, Navier-stokes equations, and Boltzmann transport equations.

Continuity Theorem
If f(x) and g(x) is continuous at point “a”,
1. The sum of the continuous function is continuous.
f(x) + f(g) is continuous at a.
2. The difference of continuous function is continuous
f(x) – f(g) is continuous at a.
3. The product of continuous function is continuous.
f(x) g(x) is continuous at a.
4. The quotient of continuous is continuous
f(x)/g(x) is continuous at a (where g(a) ≠0)
5. The functional composition of continuous function is continuous at all points x.
If f(x) is continuous at a and g(x) is continuous at f(a), then their composition g of (x) if continuous at a.
6. f(x) ≥ 0, f(x) is positive
Is continous at a.
7. Any polynomial function is continuous for all values of x.

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