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Proving a function is continuous

  • Thread starter Jeff Ford
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  • #1
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I am working to prove that this function is continuous at [itex] x = 2 [/itex]

[tex] {f(x) = 9x - 7} [/tex]

To do this I know that I have to show that [itex] \vert f(x) – f(a) \vert < \epsilon [/itex] and that [itex] \vert x - a < \delta \vert [/itex]

I tried to come up with a relationship between [itex] \vert x - 2 \vert [/itex] and [itex] \epsilon [/itex] so I could get an appropriate number to choose for [itex] \delta [/itex]

This is as far as I got

[tex] \vert {f(x) – f(a)} \vert < \epsilon [/tex]
[tex] \vert {9x – 7} \vert < \epsilon [/tex]

I’m stuck. All of the examples the text shows give equations where it is easy to factor out the [itex] \vert {x - a} \vert [/itex] term.

A push in the right direction would be appreciated.
 
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Answers and Replies

  • #2
quasar987
Science Advisor
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Wow. First of all, I suggest you use the more conventionnal sign "-" for minus, instead of 8211; :wink:.

To show a function is continuous at 'a', you must how that the limit as x approaches a is f(a). Here, a = 2 and f(2) = 9(2)-7 = 11. So given e>0, we must find 'd' such that 0<|x-2|<d ==> |9x-7 - 11|=|9x-18|=9|x-2|<e.

Mmmh.
 
  • #3
154
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Thanks. Latex was being weird yesterday. I'm not sure why it put the 8211; in there in place of some - but not all. Weird.
 

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