Page 237 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes
If but f(a) is not defined then the discontinuity at x=a can be removed by defining

f(a)=L.

Graph of (x – 1)/(x – 1) If but f(a) L
2
We can “remove” the discontinuity by filling the hole. The domain of g(x) may be extended

to include x = 1 by declaring that g(1) = 2. This makes g(x) continuous at x = 1. Since g(x) is

continuous at all other points (as evidenced, for example, by the graph), defining g(x) = 2 turns
g into a continuous function.
The limit and the value of the function are different.

If the limit as x approaches a exists and is finite and f(a) is defined but not equal to this limit, then
the graph has a hole with a point misplaced above or below the hole. This discontinuity can be

removed by re-defining the function value f(a) to be the value of the limit.
Then the discontinuity at x = a can be removed by re-defining f(a) = L.

As an example, the piecewise function in the second equipment was given by

h(x) =

Portion of the graph of h(x)

We can remove the discontinuity by re-defining the function so as to fill the hole. In this case we

re-define h(.5) = 1.5 1/(.75) = 17/6.

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