What is the difference between a limit and continuity.
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This section contains a sketch of the formal mathematics that is required to fully develop the concept of the derivative. A complete understanding is beyond the scope of this course, but a few of the ideas are sufficiently important that some discussion is warranted. In the previous sections we have discussed how the derivative is related to the slope of the tangent line for a curve at a point. This was viewed geometrically by considering a sequence of secant lines that approached the tangent line at a point or algebraically by examining what happened to the slope computed at a point as you took points closer and closer together on the curve. (Another perspective on this subject can be viewed in the University of British Columbia notes, which have had more time to be developed.) Closely connected to the concept of a limit is that of continuity. Intuititvely, the idea of a continuous function is what you would expect. If you can draw the function without lifting your pencil, then the function is continuous. Most practical examples use functions that are continuous or at most have a few points of discontinuity.
Definition: A function f(x) is continuous at a point x0 if the limit exists at x0 and is equal to f(x0).
The examples above should also help you appreciate this concept. In all of the cases except Example 3, the existence of a limit also corresponds to points of continuity. Example 3 is not continuous at x0 = 3though a limit exists here, as the function is not defined at 3. Examples 3 and 5 are discontinuous only at x0 = 3, while Examples 4, 6 and 7 are discontinuous only at x0 = 0. At all other points in the domains of these examples are continuous.
Example Comparing Limits and Continuity
An example is provided to show the differences between limits and continuity. Below is a graph of a function, f(x), that is defined on the interval [-2, 2], except at x = 0, where there is a vertical asymptote.

It is clear that the difficulties with this function occur at integer values. At x = -1, the function has the value f(-1) = 1, but it is clear that the function is not continuous nor does a limit exist at this point. At x = 0, the function is not defined (not continuous nor has any limits) as there is a vertical asymptote. At x = 1, the function has the value f(1) = 4. The function is not continuous at x = 1, but the limit does exist with

At x = 2, the function is continuous with f(2) = 3, which also means that the limit exists. At all non-integer values of x the function is continuous (hence its limit exists).
Both the geometric and the algebraic ideas mentioned above need the concept of a limit. From a conceptual point of view, the limit of a function f(x) at some point x0simply means that if your value of x is very close to the value x0, then the function f(x)stays very close to some particular value.
Definition: A function f(x) is continuous at a point x0 if the limit exists at x0 and is equal to f(x0).
The examples above should also help you appreciate this concept. In all of the cases except Example 3, the existence of a limit also corresponds to points of continuity. Example 3 is not continuous at x0 = 3though a limit exists here, as the function is not defined at 3. Examples 3 and 5 are discontinuous only at x0 = 3, while Examples 4, 6 and 7 are discontinuous only at x0 = 0. At all other points in the domains of these examples are continuous.
Example Comparing Limits and Continuity
An example is provided to show the differences between limits and continuity. Below is a graph of a function, f(x), that is defined on the interval [-2, 2], except at x = 0, where there is a vertical asymptote.

It is clear that the difficulties with this function occur at integer values. At x = -1, the function has the value f(-1) = 1, but it is clear that the function is not continuous nor does a limit exist at this point. At x = 0, the function is not defined (not continuous nor has any limits) as there is a vertical asymptote. At x = 1, the function has the value f(1) = 4. The function is not continuous at x = 1, but the limit does exist with

At x = 2, the function is continuous with f(2) = 3, which also means that the limit exists. At all non-integer values of x the function is continuous (hence its limit exists).
Both the geometric and the algebraic ideas mentioned above need the concept of a limit. From a conceptual point of view, the limit of a function f(x) at some point x0simply means that if your value of x is very close to the value x0, then the function f(x)stays very close to some particular value.
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