What on earth does this statement even mean?

[Demon117]

Homework Statement

Suppose that f : R2 |--> R and that each “slice” x l--> f(x, y), y in R, is
continuous. If df/dy (partial) exists (everywhere) and is continuous, show that f is continuous.

I don't quite understand the notation here, a little help on this will be useful. I could probably come up with what I need beyond that. I only need help defining what the first statement is saying.

 

[Skrew]

I'm pretty sure its just asking you to show that if partial derivative exists for all y and is continuous then f is continuous.

 

[Demon117]

I'm pretty sure its just asking you to show that if partial derivative exists for all y and is continuous then f is continuous.

Not exactly what I was asking. I was actually referring to the statement:

Suppose that f : R2 |--> R and that each “slice” x l--> f(x, y), y in R, is
continuous.

 

[Ryker]

The way I understand this is that for a fixed x, f(x,y) is continuous. So basically "f(y)" is continuous.

 

[bennyska]

i think it's the same: f is a mapping from R2 into R. for any fixed y, the function f(x,y) is continuous. (i think).

 

[tiny-tim]

yes …

the question would be clearer if it said "the function f_y: R -> R defined by f_y(x) = f(x,y) is continuous"