We know that for every point, there exists neighboring points.
But when we say that the function f(x) is continuous at a point x=a, we mean that at point ( a, f(a) ) the graph of the function f(x) has no holes or gaps or discontinuity. That is, its graph is unbroken at point (a, f(a))
Here say , = 2
And f(a) = 2
Then, = f(a)
Hence f(x) is continuous at x=a.
Qualitatively, the graph of a function is said to be continuous at x=a if while travelling along the graph of the function and in crossing over the point at x=a either from left to right or from right to left one does not have to lift his pen
Incase one has to lift his pen, then the graph of the function is said to have a discontinuity or break at x=a.
CONTINUITY OF A FUNCTION OF ONE VARIABLE
Function y = f(x) is said to be continuous at point x=a, where a ? domain of f(x)
If f(x) is not continuous at x=a, we say that f(x) is discontinuous at x=a, f(x) will be discontinuous at x=a if any of the following three conditions are satisfied:
i.) f(a) is not defined,
ii.) and exists but are not equal
iii.) and exists and are equal but are not equal to f(a).
- iv) At least one of the limits does not exist.
PROBLEM: Determine if the following function is continuous at x=0 .
here F(x) =
At x=0, let x = 0 + h,
ie, = = = 3.
At x = 0, let x = 0 – h,
ie, = = = 3.
But f (0) = 0
Therefore, , = = 0.
Thus, f(x) is discontinuous at x.
There are two types of discontinuities – type-1: removal discontinuity and type-2: non-removal discontinuity.
Here, necessarily exists but is either not equal to f(a) or f(a) is not defined.
In this case, therefore, it is possible to redefine the function in such a manner that and thus making the function continuous.
Here, doesn’t exist and therefore, it is not possible to redefine the function in any manner and make it continuous.