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Now that we've got the idea of a continuity at a point down, we can talk about what it means for a function to be continuous on an entire interval.
It shouldn't come as much of a surprise that we say a function f is continuous on the open interval (a,b) if f is continuous at every point c in (a,b), not including the points a and b.
In other words, f is continuous at every point in the interval (a, b).
If we try to graph a continuous function on this interval, we'll be able to draw a nice smooth curve from a to b without ever taking our pencil off the paper. We're going from Point A to Point B completely uninterrupted.
Remember, f is continuous on an interval if we can finger paint over f on that interval without lifting our drawing digit.
Look at the function f drawn below:
When we are given problems asking whether a function f is continuous on a given interval, a good strategy is to assume it isn't. Try to find values of x where f might be discontinuous.
If we're asked about the continuity of one function on several different intervals, find all the problem spots first and worry about which intervals they're in later.
If there aren't any such values in the interval, then the function is continuous on that interval.
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 and; they aren't invited. No value of x less than a or greater than b is invited, either. This is an exclusive club, with the parentheses serving as the bouncers.
In a closed interval, denoted [a, b], we're lowering our standards a bit by inviting 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'll 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
This function is continuous on the interval [a, b):
This function is continuous on (a, b) defined, is defined, and the function value at a agrees with the right-sided limit at a.
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 as well.
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).
It's all just like continuity on open intervals. The only difference is that now we have to check the endpoints.