How can you tell where on a graph f is discontinuous? What are the criteria?
usually there will be a fraction involved. Wherever the denominator is zero, f(x) will be undefined.
Or there may be a square root or logarith. f(x) not defined when the radicand is negative.
In general, look for values where f(x) is not defined; it is discontinuous there.
What if there isn't a fraction at all and it's just a graph with a function on it? My homework doesn't have a given expression of a function just a graphical representation of it.
If all you have is the graph, then if there are any vertical asymptotes or holes in the graph, that's where f(x) is discontinuous.
I mean, jeez. you must have an intuitive feel for what "continuous" means. Math usually tries to use words that already have meaning.
Alright I was clarifying since it's going to be graded. Doesn't hurt, does it?
To determine where a function is discontinuous on a graph, you need to consider three criteria:
1. Removable Discontinuity (or Point of Removable Continuity):
This occurs when there is a single point on the graph where the function has a hole or gap. To identify this type of discontinuity, check if there is any point where the function approaches a certain value from both sides but does not reach that value at the specific point.
2. Jump Discontinuity (or Point of Jump Continuity):
This occurs when the function has a sudden jump in value at a particular point. To detect a jump discontinuity, examine if there is a point where the function approaches different values from both sides.
3. Essential Discontinuity (or Point of Essential Continuity):
This occurs when there is a vertical asymptote or an infinite jump in the function. To identify an essential discontinuity, check if there is any point where the function approaches infinity or negative infinity.
By analyzing these three criteria, you can determine the locations of discontinuities on a graph. However, it is important to note that these criteria provide a general guideline, and there may be other types of discontinuities that require further analysis or mathematical techniques to identify.