Definition of Continuity
From the above discussion, the continuity with reference to a function may be defined as under:
“A function is said to be continuous at a point, x =a, if f(x) possesses a finite, or a definite limit as x tends to the value, a from either side (lower or upper side of a, say 3 viz. 2.9, 2.99 etc. or 3.1, 3.01 etc. respectively), and each of these limits equal to f(a)”.
From the above definition, the essential characteristics of a continuous function may be laid down as follows :
i. f is defined at a
ii. lim/x→a f(x) = f(a)
iii. f(a), lim f(x) and lim f(x) all have finite and definite values.
iv. The values of lim f(x) , f(a) and lim f(x) are identical so that lim f(a-h) = f(a) =lim/h→0 f(a+ h)
If , any of the above essential characteristics (known as conditions) is not satisfied then the function will be deemed as discontinuous at the given point In other words, the given point will be a point of discontinuity of the function.