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- Mar 10th 2010, 10:00 PM #1
## Contuinity

I am confused with some true or false reasoning questions on continuity.

a. If the function f+g:R -> R is continuous, then the functions f:R -> R and g:R -> R are also continuous.

b. if the function :R->R is continuous, so is the function f:R->R

c. If the function f+g:R -> R and g:R -> R are continuous, then f:R -> R is also continuous.

d. Every function f:N -> R is continuous, where N denotes the set of natural numbers.

I think that (a) and (c) are true by the definition of continuity. If either f:R -> R or g:R -> R, then according to the definition of continuity, f+g cannot be continuous.

I am unsure about how (b) and (d) are to be done. Any suggestions?

- Mar 10th 2010, 10:49 PM #2

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- Mar 11th 2010, 04:55 AM #3

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Look at f(x)= -1 if x is rational, 1 if x is irrational.

c. If the function f+g:R -> R and g:R -> R are continuous, then f:R -> R is also continuous.

d. Every function f:N -> R is continuous, where N denotes the set of natural numbers.**only**sequence that converges to is the constant seqence {a, a, a, a, ...}.

I think that (a) and (c) are true by the definition of continuity. If either f:R -> R or g:R -> R, then according to the definition of continuity, f+g cannot be continuous.

I am unsure about how (b) and (d) are to be done. Any suggestions?