2.5 Continuity Informally, a continuous function is a function whose graph has no breaks, holes, or vertical asymptotes. To illustrate, the graph of each function shown below is not continuous at the number 1 . i) ii) iii) Function i) has a removable discontinuity at 1 , function ii) has a jump discontinuity at 1 , and function iii) has an infinite discontinuity at 1 . Definition: A function f is continuous at a number c if the following conditions are satisfied: a.
f (c ) is defined b.
lim f  x  exists c.
lim f  x   f (c) x c
x c
Whenever this definition is used to show that a function is continuous at c , it is sufficient to verify only the third condition , because if lim f  x   f (c) , then the first two conditions are satisfied x c
automatically. In the previous illustration, function i) has a removable discontinuity at 1 because the discontinuity could be removed by defining the function value f ( 1) appropriately.