What is the difference between these two statements?

T

The Student

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My notes has the follow: "Corollary 3.1.7 (Composition of Continuous Functions): Suppose g is continuous at a and f is continuous at g(a). Then f ◦g is continuous at a". What is the difference between "f is continuous at g(a)" and "f ◦g is continuous at a"? I thought that the former meant the latter? In other words, isn't f at g(a) the same as f ◦g at a? If not, what is the difference?
 
The Student said:
My notes has the follow: "Corollary 3.1.7 (Composition of Continuous Functions): Suppose g is continuous at a and f is continuous at g(a). Then f ◦g is continuous at a". What is the difference between "f is continuous at g(a)" and "f ◦g is continuous at a"? I thought that the former meant the latter? In other words, isn't f at g(a) the same as f ◦g at a? If not, what is the difference?
Click to expand...

There is no difference between the two statements. That's the point of the corollary.
 
Last edited:
fcabanski said:
There is no difference between the two statements. That's the point of the corollary.
Click to expand...

There is a difference. Take for example

\(\displaystyle g(x) = \begin{cases}-1 & \text{if } x < 0 \\
1 & \text{if } x \ge 0
\end{cases}\)

and

\(\displaystyle f(x)=x\).

Then \(\displaystyle f\) is continuous at \(\displaystyle g(0)=1\). But \(\displaystyle f\circ g \) (=g) is not continuous at zero, since \(\displaystyle g\) is not continuous at zero.
 
daon2 said:
There is a difference. Take for example

\(\displaystyle g(x) = \begin{cases}-1 & \text{if } x < 0 \\
1 & \text{if } x \ge 0
\end{cases}\)

and

\(\displaystyle f(x)=x\).

Then \(\displaystyle f\) is continuous at \(\displaystyle g(0)=1\). But \(\displaystyle f\circ g \) (=g) is not continuous at zero, since \(\displaystyle g\) is not continuous at zero.
Click to expand...
When f is used by itself, is it the same as using f(x) in this instance?
 
I was referring to the entire statement. "Suppose g is continuous at a and f is continuous at g(a). Then f ◦g is continuous at a"." As far as continuity, the two situations are the same. That's what the corollary is stating.

I see I misunderstood the question. "f is continuous at g(a)" and "f ◦g is continuous at a" - when are they different, or how could they be different, was the question.
 
The point you are missing is that the two statements are NOT "the same". "f is continuous at g(a) and g is continuous at a" implies that "f ◦g is continuous at a". But "f ◦g is continuous at a" does NOT imply "f is continuous at g(a) and g is continuous at a". They are NOT interchangeable. That was the point of daon2's example.
 
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