Composite function and continuerty

[Mathman23]

Hello I have the following problem:

Given two function f and g which are continuer on R, and some point c which belongs to R.

I'm suppose to show that if f(c) = g(c), then h is continious on R.

Isn't that the same as showing that that the composite function

h(c) = g(f(c)) is continues on R?

Best Regards,

Fred

 

[Muzza]

Given two function f and g which are continuer on R, and some point c which belongs to R.

I'm suppose to show that if f(c) = g(c), then h is continious on R.

You haven't told us what "h" is.

 

[Mathman23]

You haven't told us what "h" is.

h is definied as follows

$$\begin{displaymath} h(c) = \left\{ \begin{array}{ll} f(c) \\ g(c) \\ \end{array} \right. \end{displaymath}$$


$$c \in \mathbb{R}$$ and f and g er continious on R.

Then how do I show that if f(c) = g(c), then h(c) is continious on R?

My own ideer is to show this if f and g are continious on R, then the composite function h is continious on R.

Best Regards

/Fred

 

[Muzza]

I'm afraid I have no idea what that means.

 

[Mathman23]

Okay let's look it at another way,

f and g: $$[a,b] \rightarrow \mathbb{R}$$ are continious, $$c \in ]a,b[$$. Next, let h be

$$\begin{displaymath}h(x) = \left\{ \begin{array}{ll}f(x) \ \ \ \\g(x) \\\end{array} \right.\end{displaymath}$$

h is defined on [a,b]

Now my assigment is the following:

a) Let f and g be arbitrary functions. Show that if f(c) = g(c), then the function h is continious. I sure I need to use the epsilon-delta definition of continuerty, but if there is anybody out there who maybe can explain what I need to do by way of an example I would very much appreciate it :-)

b) If $$f(c) \neq g(c)$$ the h is discontinious. Anybody who can direct me to a good example on how to show this?

Sincerely and God bless You :-)

Fred

/Fred