Give an example of a sequence of continuous functions $f_n:\mathbb{R} \to \mathbb{R}$ such that the pointwise limit $f:\mathbb{R} \to \mathbb{R}$ exists but is discontinuous.
My idea is $f_n(x)=\displaystyle\frac{x}{n}.$ Then the pointwise limit is $0$ but $f$ is discontinuous on $\mathbb{R}.$ Is that correct?
I'm having a hard time understanding the definition of pointwise convergence. I have this definition:
$\displaystyle\lim_{n\to \infty} f_n=f$ pointwise iff $\displaystyle\lim_{n\to \infty} f_n(x)=f(x)$
But then what is $f(x)$? Where does the $n$ go? In the example I gave, for instance,
$\displaystyle\lim_{n\to \infty} f_n(x)=\displaystyle\lim_{n\to \infty} \frac{x}{n}=0.$ But does this equal $f(x)$? Is $f(x)$=0 a discontinuous limit?
Best Answer
The example you give has a point-wise limit of 0, which is not a discontinuous function. Some standard examples of these kinds of functions are ones that kind of smoothly interpolate between two values, with the interpolating region getting smaller and smaller. For example, consider the sequence:
$$ f_n(x) = \tan^{-1}(nx) $$
This is plotted for some various values of $n$ below, on the region $x\in [-1,1]$
Along with Suugaku's answer, this should give you the intuition you need about point-wise limits and continuity.