There are no holes, no jumps, no asymptotes, or any other form of discontinuity.Ĭontinuity is formally defined using limits. It is possible to trace the entire function without the need to raise the pencil. On the other hand, the figure below is an example of a continuous function: Referencing the figure above, at the point (3, 6), one would have to lift their pencil to draw the graph, so it is discontinuous. One way to determine whether or not the graph of the function is continuous is to attempt to draw/trace the function without needing to lift the pencil if it can be drawn without lifting the pencil, the function is continuous otherwise, the function is discontinuous.
Informally, a function is said to be continuous if its graph is a single unbroken curve with no holes. Continuityįunctions can be either continuous or discontinuous. The function above is said to be discontinuous at x = 3, depicted as the open circle in the figure above. It also provides the means for us to discuss another far-reaching concept in calculus, that of continuity. Limits allow us to describe the behavior of the function at x = 3 and state that the function approaches 6 even though the function is undefined at that point. The graph of the function is shown in the figure below:Īs we determined above, the function is undefined at the point x = 3. In this case, it is possible to evaluate the limit using the third option, by factoring the numerator, thereby allowing us to cancel the denominator: There are a number of different ways to evaluate the limit of a function at a given point, including graphically, numerically, or in some cases, by simply evaluating the limit at the given point. In other words, rather than evaluating the expression at x = 3, we find the limit as x approaches 3. The concept of limits allows us to study the behavior of the function as x gets closer and closer to a given point (in this case 3), even though we cannot evaluate it at exactly that point. If we evaluate the expression at x = 3, we find that it is undefined, since the x - 3 in the denominator would evaluate to 0. It provides information about a function's behavior near a point, rather than exactly at that point, which is important since determining the behavior of a function at a specific point is not always possible.įor example, consider the expression. LimitsĪ limit is the value that a function approaches as its input value approaches some value. The concepts of limits and continuity form the foundation of the study of calculus. Uniform continuity looks good.Home / calculus / limits and continuity Limits and continuity
Bounded is insufficient but bounded derivative probably works.
Lipschitz continuous, differentiable, and even smooth are insufficient. We can probably find a different condition, but those two counterexamples rule out lots of good tries. You should be able to see the contradiction and it would just need to be formalized. If you don't see why this is a problem, draw it. $\ \lim\limits_.$$ One plan for showing this is continuous is by contradiction suppose there was an $\varepsilon$ such that for every $\delta$ there is some a $x\in(b-\delta,b]$ such that $f(x)\notin (f(b-\delta),f(b))$. Recall the 3-part definition of "$f(x)$ is continuous at $x=a$" from elementary calculus:Ģ.
Others have already answered, but perhaps it would be useful to have at least one of the answers target the elementary calculus level.
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