Proving Continuity of a Piecewise Function at x=1

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Homework Help Overview

The problem involves analyzing the continuity of a piecewise function defined at a specific point, x=1. The function is given in two parts, one for x less than 1 and another for x greater than or equal to 1.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the values of the function at x=1 and question whether they are equal. There is an attempt to understand how to prove continuity mathematically, alongside concerns about the implications of continuity for subsequent parts of the problem.

Discussion Status

The discussion is ongoing, with participants expressing uncertainty about the continuity of the function. Some have suggested that the values at x=1 are not equal, while others are exploring the implications of this assumption for the overall problem.

Contextual Notes

There is mention of the need for more decimal precision in evaluating the function values, as well as the algebraic independence of the constants involved, which may affect the continuity assessment.

chaoseverlasting
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Homework Statement


f(x) is a piecewise function defined as:

[tex]2(2e-e^x) x<1[/tex]
[tex]3\pi x-4 x>=1[/tex]

Discuss the continuity of f(x) at x=1.


Homework Equations





The Attempt at a Solution



Putting x=1 in the above function gives you 2e and [tex]3\pi -4[/tex]. They seem to be equal, but how do I prove it mathematically? I've missed something here, but don't know what.
 
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How many decimal places are you looking at? They aren't THAT close to being equal.
 
No idea. Is this thing continuous at x=1?
 
chaoseverlasting said:
No idea. Is this thing continuous at x=1?

It's a lot easier to show they aren't equal than that they are. Hint: they can't be equal, e and pi are algebraically independent. Just punch the things into calculator that shows more than two decimal places.
 
Last edited:
Yeah, I did just that. The question has other parts which can only be solved if this thing was continuous. Nasty assumption.
 
2e and 3[itex]\pi[/itex]-4 are definitely NOT equal!
 

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