What on earth does this statement even mean?

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Homework Help Overview

The problem involves a function f mapping from R² to R, with a focus on the continuity of its "slices" for fixed values of y. The original poster seeks clarification on the notation and meaning of the statement regarding the continuity of these slices and the implications for the overall continuity of f.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the meaning of the notation and the implications of continuity for fixed values of y. Some express uncertainty about the interpretation of the statement regarding slices and continuity.

Discussion Status

Several participants are exploring the meaning of the initial statement, with some suggesting that it implies continuity for fixed x values. There is a shared effort to clarify the notation and its implications without reaching a consensus on the interpretation.

Contextual Notes

Participants note that the original statement could be clearer, particularly regarding the definition of the function for fixed y values. There is an acknowledgment of the potential ambiguity in the notation used.

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

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

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