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Continuity of Functions

Continuity of Functions

Continuity on Closed and Half-Closed Intervals

When looking at continuity on an open interval, we only care about the function values within that interval.

If we're looking at the continuity of a function on the open interval (a,b), we don't include a at b, they aren't invited. No value of x less than a or greater than b is invited, either. This is an exclusive club.

In a closed interval, denoted [a,b], we also must also invite our friends a and b to the pool party. Half-closed intervals either invite a, [a,b), or b, (a,b]. To talk about continuity on closed or half-closed intervals, we will see what this means from a continuity perspective. Start with a half-closed interval of the form [a,b). What does it mean for a function to be continuous on this interval?

Since we can only approach a from the right, we use the continuity definition for a right-sided limit instead of a two sided limit. We say f is continuous on [a,b) if f is continuous on (a,b) and

  • f(a) exists
      
  •  exists
      
  • f(a) and  agree

Sample Problem

This function is continuous on the interval [a,b):

This function is continuous on (a,b). f(a) is defined,  is defined, and the function value at a agrees with the right-sided limit at a.

Sample Problem

The following functions are not continuous on the interval [a,b).

This function is not continuous on [a,b) because f(a) is undefined. We could also say this function is not continuous on [a,b) because  does not exist.

This function is not continuous on the interval [a,b) because .

This function is not continuous on the interval [a,b) because it is not continuous on the open interval (a,b).

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