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 $f^2$: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?

2. Originally Posted by harish21
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 $f^2$: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?
a) Consider functions on $\mathbb{R}$:

$f(x)=\begin{cases}0,&x<0\\1,& x\ge 0 \end{cases}$

$g(x)=\begin{cases}1,&x<0\\0,& x\ge 0 \end{cases}$

$h(x)=f(x)+g(x)=1$ and so is continuous on $\mathbb{R}$

So we conclude ...

CB

3. Originally Posted by harish21
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 $f^2$:R->R is continuous, so is the function f:R->R
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.
A function, f(x), is continuous at x= a if and only if $f(x_n)\to f(a)$ for any sequence $x_n\to a$. If the domain of f is N, the only sequence that converges to $a\in N$ 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?