The definition of continuity was framed after decades of deliberation and mathematical squabbling. The current notation we have is due to a Polish mathematician by the name of Weierstrass. It states that
“If is continuous at point , then for every , such that for , .”
Now let us try and interpret the statement and break it down into simpler statements, in order to give us a strong visual feel.
Can be very large? Of course! It can be for example. Does there exist a such that , even if the function is not continuous? Yes. An example would be
for and for
Does this mean that we have proved a discontinuous function to be continuous? NO.
should take up the values of all positive real numbers. So defined above will fail for lower than
Let us suppose for some , we have if . Let and be two points in . Let us now make . Will the value of also have to decrease? Can it in fact increase?
The value of cannot increase because the bigger interval will contain and , and we know that that will violate the condition that for all points in , the distance between the mappings has to be less than . Can remain the same? No (for the same reasons, as the interval will still contain and ). Hence, most definitely has to decrease in this case?
However, does it always have to decrease? No. An example in case is a constant function like .
We have now come to the most important aspect of continuity. The smaller we make , the smaller the value of . Does continuity also imply that the smaller we make , the smaller the value of ? YES! How? When we decrease , obviously can’t get bigger. Moreover, we know that there do exist values of which make smaller possible. Say, for , it is necessary that . Hence, if we decrease the radius of the interval on the x-axis from to , the value of (or the bound of the mappings of the points) also decreases to .
In summation, a continuous function is such that
decrease in value of decrease in value of
One may ask how does knowing this help?
It has become very easy to prove that differentiable functions are continuous, and a host of other properties of continuous functions.
A doubt that one may face here is does this imply that all continuous functions are differentiable? No. “decrease in value of decrease in value of ” just implies that the derivative formula at will have a limit for every cauchy sequence of converging to . In order for a function to be derivable, all those limits of the different cauchy sequences have to be equal. This is not implied by the aforementioned condition.