Statistics

f(x)\) is continuous.



......(i)
Now for





....(ii)
Adding (i) and (ii) we get

Hence, is continuous for

The ability to trace a function's graph with a pencil without taking the pencil off the paper is a feature of many functions. These are referred to as continuous functions. If a function's graph does not break at a particular point, it is said to be continuous at that location. In general, an introductory calculus course will give a precise explanation of how the limit concept applies to the continuity of a real function. First, a function f with variable x is continuous at point "a" on the real line if and only if the limit of f(x) equals the value of f(x) at "a," i.e., f(a), as x approaches "a."

Following are some mathematical definitions of continuity:

If the following three conditions are met, a function is said to be continuous at a given point.

• f(a) is defined
• lim _x→a f(x) exists
• lim_x→a f(x)=lim _x→a f(x)=f(a)

When a graph can be traced without lifting the pen from the sheet, the function is said to be a continuous function. A function, on the other hand, is said to be discontinuous if it contains any gaps in between.