ANALYSIS II: continuity of function in R^n

In summary, the conversation discussed the continuity of a function in n-dimensional space. It was shown that if the partial derivatives of the function exist and are bounded on a subset of the space, then the function is continuous at a specific point. The conversation also explored ways to prove this using Lipschitz continuity and estimating the difference between the function at a point and the function at a sequence of points approaching that point.
  • #1
jjou
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[SOLVED] ANALYSIS II: continuity of function in R^n

Let [tex]A\subset\mathbb{R}^n[/tex] and [tex]f:A\rightarrow\mathbb{R}[/tex]. Show that, if the partial derivatives [tex]D_jf(x)[/tex] exist and are bounded on [tex]B_r(a)\subset A[/tex], then f is continuous at a.

We know:
[tex]D_jf(x_1,...,x_n)=\lim_{h\rightarrow0}\frac{f(x_1,...,x_j+h,...,x_n)-f(x_1,...,x_n)}{h} \leq B[/tex] for all [tex]1\leq j\leq n[/tex], |h|<r for some B>0.
f is continuous at a if, for any [tex]\epsilon>0[/tex], there exists [tex]\delta>0[/tex] such that [tex]|f(x)-f(a)|<\epsilon[/tex] whenever [tex]||x-a||<\delta[/tex].
Equivalently, f is continuous at a if, for any sequence [tex](x_n)[/tex] converging to a, [tex]f(x_n)[/tex] converges to f(a).

I tried showing f to be Lipschitz (and thus continuous) or to estimate |f(x)-f(a)| for [tex]x\in B_r(a)[/tex] by the following:
[tex]|f(x)-f(a)|\leq|f(x)-f(x_1,...,x_{n-1},a_n)|+|f(x_1,...,x_{n-1},a_n)-f(x_1,...,x_{n-2},a_{n-1},a_n)|+...+|f(x_1,a_2,...a_n)-f(a)|[/tex]
since then each term in the right hand sum involves only a change in one coordinate (closer to a partial derivative).

I'm getting stuck though on how to proceed from here since partials involve taking limits, whereas continuity does not. Also, the existence of partials does not guarantee differentiability of f. Am I on the right track or way off? Any hints would be highly appreciated.

Thank you. :)
 
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  • #2
Can anyone help please?
 
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  • #3
Got it. :)
 

1. What is the definition of continuity for a function in R^n?

The definition of continuity for a function in R^n is that the function must have the same limit at a point a as it approaches from any direction.

2. How is continuity of a function in R^n different from continuity in R?

In R, a function is continuous if the limit of the function at a point is equal to the value of the function at that point. In R^n, a function is continuous if the limit of the function at a point is equal to the value of the function at that point regardless of which direction it is approached from.

3. How can I determine if a function in R^n is continuous?

A function in R^n is continuous if it is continuous at every point in its domain. This can be determined by checking if the limit of the function at a point is equal to the value of the function at that point from all directions.

4. What is the importance of continuity in R^n?

Continuity in R^n is important because it allows us to make predictions and approximations about the behavior of a function at a point based on its behavior in the surrounding points. It also allows us to apply mathematical concepts such as derivatives and integrals to these functions.

5. How does continuity of a function in R^n relate to differentiability?

A function in R^n is differentiable at a point if it is both continuous at that point and has a well-defined tangent plane at that point. Therefore, continuity is a necessary condition for differentiability in R^n.

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