his page is a sub-page of our page on Basic properties of functions.
///////
The sub-pages of this page are:
• …
///////
Related KMR-pages:
• …
///////
Other relevant sources of information:
• Norm
• Topology
• Metric
• Pointwise convergence
• Uniform convergence
• Almost everywhere convergence
///////
The interactive simulations on this page can be navigated with the Free Viewer
of the Graphing Calculator.
///////
Uniform convergence:
///
The interactive simulation that created this movie.
Pointwise – but not uniform – convergence:
The interactive simulation that created this movie.
When \, n \rightarrow \infty \, this sequence converges pointwise for each value of \, x , i.e., \, f_n(x) \rightarrow F(x) \, for each \, x . However, the convergence is not uniform, since there is no ”tail-value” \, N = N(\epsilon) \, such that the ”tail” of the sequence \, f_n(x) \, stays within \, \epsilon \, of the pointwise limit function \, F(x) \, for EVERY value of \, n that is greater than \, N . In order for the convergence to be uniform, the tail of the sequence \, f_n(x) \, must stay within the epsilon-band of the limit function \, F(x) \, FOR EVERY VALUE of \, \epsilon > 0 .
The disappearing wave:
Let \, g_n(x) = \dfrac{nx}{e^{nx}} \, be given by the red curve and consider the sequence \, { \{ g_n \} }_{n=1}^{\infty} .
Each function \, g_n \, is continuous at the point \, x = 0 ,
since when \, x \rightarrow 0 \, we have \, \lim\limits_{x \to 0} g_n(x) = 0 = g_n(0) \, for each \, n \, .
When \, x = 1/n \, we have \, g_n(1/n) = 1/e , which is the maximum value of \, g_n .
The sequence of functions \, { \{ g_n \} }_{n=1}^{\infty} behaves like a wave that ”compresses itself” towards the point \, x = 0 \, and threatens to break at this point. Each member function \, g_n \, attains its maximum amplitude of \, 1/e \, at the point \, x = 1/n . The “sequence-wave” passes by each point \, x > 0 \, and then ”dies down” towards amplitude \, 0 \, at this point. Yet the wave never reaches the point \, x = 0 , because at this point it always has the amplitude \, 0 \, since \, g_n(0) = 0 \, for each value of \, n .
//////