# Thread: finding a sequence of continuous bounded functions

1. ## finding a sequence of continuous bounded functions

I want to find a sequence (f_k)of continuous bounded functions on [0,1]

converging pointwise to a continuous limit, but such that

the sup-norm of f_k tends to infinity when k tends to infinity...

is there any?

2. Nope, i don't think there are any. Since if $f_k \rightarrow f$ pointwise then
$\lVert f_k \lVert_{sup} \rightarrow \lVert f \lVert_{sup}$ which would mean f is not bounded on [0, 1] (since $\lVert f_k \lVert_{sup} \rightarrow \infty$ ) thus it must not continuous on [0, 1].

3. Originally Posted by losin90
I want to find a sequence (f_k)of continuous bounded functions on [0,1]

converging pointwise to a continuous limit, but such that

the sup-norm of f_k tends to infinity when k tends to infinity...

is there any?
What you need to do is to ensure that the function $f_k$ has a narrow spike somewhere. For example, define $f_k(x) = k^2x$ in the interval $0\leqslant x<1/k$, $f_k(x) = \tfrac2k - k^2x$ in the interval $1/k\leqslant x<2/k$, and $f_k(x) = 0$ in the remainder of the unit interval. Then $f_k(x)\to0$ as $k\to\infty$, for each (fixed) point x in the unit interval.

Originally Posted by jakncoke
Nope, i don't think there are any. Since if $f_k \rightarrow f$ pointwise then
$\lVert f_k \lVert_{sup} \rightarrow \lVert f \lVert_{sup}$ which would mean f is not bounded on [0, 1] (since $\lVert f_k \lVert_{sup} \rightarrow \infty$ ) thus it must not continuous on [0, 1].
That is not true, because pointwise convergence does not imply convergence in the sup norm.