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Continuity of Functions
Function 
is continuous at point 
if the following three conditions are satisfied:
i.) 
is defined,
ii.) 
exists (i.e., is finite),
iii.) 
.
Function f is said to be continuous on an interval I if f is continuous at each point x in I. Here is a list of some well-known facts related to continuity:
1. The sum of continuous functions is continuous.
2. The difference of continuous functions is continuous.
3. The product of continuous functions is continuous.
4. The quotient of continuous functions is continuous at all points x where the denominator is not zero.
5. The functional composition of continuous functions is continuous at all points x where the composition is properly defined.
6. Any polynomial is continuous for all values of x.
7. Function ex and trigonometry functions 
and 
are continuous for all values of x.
f (x) is not continuous at x = a, then f (x) is said to be discontinuous at this point. Figure 1 shows the graphs of four functions, two of which are continuous at x = a and two are not.
| 
| |
| Continuous at x = a. | Discontinuous at x = a. | |
| 
| |
| Continuous at x = a. | Discontinuous at x = a. | |
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