Continuity
The next important fundamental concept is continuity or determining if a function is continuous.
In order for a function to be continuous at a particular point, it must first have a limit at the point - if it does not have a limit, it can not be continuous. Secondly, the value of the function must also equal the limit. More formally, lim x->A+ f(x) = lim x->A- f(x) and = f(A). In other words, there cannot be any breaks, jumps or gaps in the function or graph.
More informally, a simple way to conceptually understand the notion of continuity is that when drawing or tracing the graph, you never lift your pencil when a function is continuous. If you lift your pencil, it is not a continuous function. Please note that a function can be continuous for all values on an interval, except for a single point.
Special note: The "lifting your pencil test" is not an acceptable means of proving continuity, just an aide for conceptually understanding continuity. For showing continuity at a point B on the function f(x), you must show that lim x->B+ f(x) = lim x->B- f(x) and = f(B) or you can simply say there is a break in the graph or in another case an asymptote exists exists and it is not continuous.
In order for a function to be continuous at a particular point, it must first have a limit at the point - if it does not have a limit, it can not be continuous. Secondly, the value of the function must also equal the limit. More formally, lim x->A+ f(x) = lim x->A- f(x) and = f(A). In other words, there cannot be any breaks, jumps or gaps in the function or graph.
More informally, a simple way to conceptually understand the notion of continuity is that when drawing or tracing the graph, you never lift your pencil when a function is continuous. If you lift your pencil, it is not a continuous function. Please note that a function can be continuous for all values on an interval, except for a single point.
Special note: The "lifting your pencil test" is not an acceptable means of proving continuity, just an aide for conceptually understanding continuity. For showing continuity at a point B on the function f(x), you must show that lim x->B+ f(x) = lim x->B- f(x) and = f(B) or you can simply say there is a break in the graph or in another case an asymptote exists exists and it is not continuous.
Determining Continuity from a Graph
Determining Continuity from a Table
Determining Continuity Algebraically for a Piecewise Function I
Determining Continuity Algebraically for a Piecewise Function II
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