Math Help - General definition of uniform convergence

1. General definition of uniform convergence

The common definition of uniform convergence is based on sequence, that is, the functions are indexed by natural numbers. But what if the functions are indexed by real number, e.g. $f_t\to f$ uniformly as $t\to t_0$? I guess the definition might be: for any $\epsilon>0$, there is a $\delta>0$ such that $|f_t(x)-f(x)|<\epsilon$ for all $x$ whenever $0<|t-t_0|<\delta$. My question is: Is it right to exclude the point $t_0$, like we do for ordinary definition of limits?
In addition, Does this type of uniform convergence preserves properties like differential, integral and limits of functions, as does the common case? Thanks!

2. Originally Posted by zzzhhh
The common definition of uniform convergence is based on sequence, that is, the functions are indexed by natural numbers. But what if the functions are indexed by real number, e.g. $f_t\to f$ uniformly as $t\to t_0$? I guess the definition might be: for any $\epsilon>0$, there is a $\delta>0$ such that $|f_t(x)-f(x)|<\epsilon$ for all $x$ whenever $0<|t-t_0|<\delta$. My question is: Is it right to exclude the point $t_0$, like we do for ordinary definition of limits?
In addition, Does this type of uniform convergence preserves properties like differential, integral and limits of functions, as does the common case? Thanks!
First, in the ordinary definition of limits, we don't exclude $t_0$. The reason why some people do so is usually because they are embarrassed with limits like $\lim_{x\to 0}\frac{\sin x}{x}=1$. The correct way to justify this limit is to write

$\lim_{x\to 0,\ x\neq 0} \frac{\sin x}{x} =1$

and read "the limit of $\frac{\sin x}{x}$ when $x$ tends to $0$ in $\mathbb{R}\setminus\{0\}$ is 1". It is just because of lazyness, or because the function is obviously not defined at the limit point, that we usually skip the " $x\neq 0$" part.

As for the uniform convergence of functions, you can use the "sequential characterization of limits" to prove everything you suggest: $f_t$ converges uniformly to $f$ as $t\to t_0$ if, and only if, for any sequence $(t_n)_{n\geq 0}$ converging to $t_0$, the sequence $(f_{t_n})_{n\geq 0}$ converges uniformly to $f$. Thus you reduce to a sequence and can apply the usual properties. For instance (under usual hypotheses, like $f_t$ continuous), for any sequence $(t_n)_{n\geq 0}$ converging to $t_0$, $f_{t_n}$ converges uniformly to $f$, hence we have $\int_0^1 f_{t_n}(x) dx \to_n \int_0^1 f(x)dx$. And using the sequential characterization (in a different way), you deduce $\int_0^1 f_t (x) dx\to_{t\to t_0} \int_0^1 f(x)dx$.

3. Originally Posted by Laurent
First, in the ordinary definition of limits, we don't exclude $t_0$. The reason why some people do so is usually because they are embarrassed with limits like $\lim_{x\to 0}\frac{\sin x}{x}=1$. The correct way to justify this limit is to write

$\lim_{x\to 0,\ x\neq 0} \frac{\sin x}{x} =1$

and read "the limit of $\frac{\sin x}{x}$ when $x$ tends to $0$ in $\mathbb{R}\setminus\{0\}$ is 1". It is just because of lazyness, or because the function is obviously not defined at the limit point, that we usually skip the " $x\neq 0$" part.
I have to disagree. Your definition of the limit of a function may be the norm in France, but it just might be a minority view outside of France. Take for example, the definition that Wikipedia offers.
Of course, I agree that Wikipedia is certainly not the highest authority when it comes to definitions of basic mathematical terms. But similar definitions can be found in many books, like for example, "The Principles of Mathematical Analysis" by Walter Rudin. Many German authors agree with that, too. (Although there are exceptions. Maybe it depends on whether one follows Bourbaki? - Could be: I don't know where, exactly, that divergence in mathematical notation and terminology comes from, but it certainly exists.)

4. Originally Posted by Failure
I have to disagree. Your definition of the limit of a function may be the norm in France, but it just might be a minority view outside of France. Take for example, the definition that Wikipedia offers.
Of course, I agree that Wikipedia is certainly not the highest authority when it comes to definitions of basic mathematical terms. But similar definitions can be found in many books, like for example, "The Principles of Mathematical Analysis" by Walter Rudin. Many German authors agree with that, too. (Although there are exceptions. Maybe it depends on whether one follows Bourbaki? - Could be: I don't know where, exactly, that divergence in mathematical notation and terminology comes from, but it certainly exists.)
You seem to be right, thanks for mentioning it, I didn't suspect this.

I just checked: the French wikipedia gives my definition, and a footnote (+ a discussion) confirms what you wrote.

Your definition dates back to Weierstrass. The French one matches Bourbaki's definition : the set of the neighbourhoods of $t_0$ is a filter, while the set of the neighbourhoods without $t_0$ is not; therefore Weierstrass' definition was not suitable for a natural generalization. So I guess that's because of filters (which nobody teaches anymore) that at some point in history we chose a different definition. Or maybe it was before, like from Cauchy? That's a question I may ask a historian of mathematics I know.