04-02-2013, 09:26 AM
Perfect Maths - II
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Continuity of a function at a point
From geometrical considerations we derive the concept of continuity. If the graph of function y = f(x) is a
continuous curve, we naturally call the function a continuous one, it means there should not be any sudden change
in the value of the function i.e. the small change in the value of ‘x’ should produce a small change in the value of
‘y’ and so the graph of function should be a continuous curve without any break in it, but the analytical definition
of continuity is quite distinct from geometrical concept of the term.
So before arriving at the formal definition of continuity, let us consider the function y = f(x) which is defined in
any closed interval [a, b]. Let x = c, be any point in this interval. Then the function f(x) varies continuously at x =
c, if the change in any value of the function f(x) is small, and if the change in the value of ‘x’ (when x varies from
x = c on either side of it) is also small.
Let us consider the value c + h of ‘x’ in the interval [a, b] where ‘h’ (the change in the value of the independent
variable ‘x’) may be positive or negative. Then the corresponding change in the value of the function f(x) or
change in the dependent variable ‘y’ is f(c + h) − f© which also may be positive or negative. Now if the function
f(x) is continuous then the value of |f(c + h) − f©| should be small if |h| is small, as we like be choosing |h|
sufficiently small. But, here the word small is some what indefinite and so cannot be the part of definition.