Making sense of continuity at a point where f(x) = Infinity?

In summary, the conversation discusses the concept of continuity for an extended real-valued function at a point where the function's value is infinite. The standard definition of continuity breaks down in this case since it is impossible to make the absolute value of the difference between infinity and any real number less than a given epsilon. However, one possible way to make sense of continuity in this scenario is by considering the reciprocal of the function's value at the point of interest, and using this to redefine the condition for continuity.
  • #1
AxiomOfChoice
533
1
Is there a way to make sense of the following statement: "[itex]f[/itex] is continuous at a point [itex]x_0[/itex] such that [itex]f(x_0) = \infty[/itex]?" The standard definition of continuity seems to break down here: For any [itex]\epsilon > 0[/itex], there is no way to make [itex]|f(x_0) - f(x)| < \epsilon[/itex], since this is equivalent to making [itex]|\infty - f(x)| < \epsilon[/itex], which cannot happen, since [itex]\infty - y = \infty[/itex] for every [itex]y\in \mathbb R[/itex] and [itex]\infty - \infty[/itex] is undefined. So is there any way to make sense of continuity of an extended real-valued function at a point where it's infinite?
 
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  • #2
Could you try regarding [itex]f(x_0) = \infty[/itex] as [itex]1/f(x_0) = 0[/itex], and so make

[tex]\left|\frac{1}{f(x_0)}-\frac{1}{f(x)} \right| < \epsilon[/tex]
?
 

FAQ: Making sense of continuity at a point where f(x) = Infinity?

What does it mean for f(x) to equal infinity at a specific point?

When f(x) equals infinity at a specific point, it means that as x approaches that point, the value of f(x) becomes infinitely large. This can happen when the function has a vertical asymptote at that point, meaning that the function approaches infinity as x approaches that value.

How can we make sense of continuity at a point where f(x) = infinity?

Continuity at a point where f(x) equals infinity can be understood by looking at the behavior of the function as x approaches that point. If the function has a vertical asymptote at that point, it is not considered continuous because there is a break in the graph. However, if the function approaches infinity from both sides of the vertical asymptote, it is considered to be continuous.

Can a function have multiple points where f(x) = infinity?

Yes, a function can have multiple points where f(x) equals infinity. This can happen if the function has multiple vertical asymptotes, or if it has a vertical asymptote at a repeating value of x, such as in the case of a trigonometric function.

How can we graph a function with a point where f(x) = infinity?

To graph a function with a point where f(x) equals infinity, we can plot the points on either side of the vertical asymptote and then draw a line to show the behavior of the function as x approaches the point where f(x) equals infinity. We can also use a graphing calculator to plot the function and see how it behaves near the point where f(x) equals infinity.

How does the concept of infinity relate to limits in calculus?

The concept of infinity is closely related to limits in calculus. A limit is used to describe the behavior of a function as the input (x) approaches a specific value. When the limit approaches infinity, it means that as x gets closer and closer to that value, the output (f(x)) becomes infinitely large. Limits are essential in understanding continuity and the behavior of functions at points where f(x) equals infinity.

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