Consider functions $ f_n (x) = \frac{1}{1+n^2 x^2} $.
Pointwise limit $ \displaystyle f(x) = \lim_{n\to\infty} f_n (x) $ is $ 1 $ at $ 0 $ and $ 0 $ at every non-zero $ x $ (esp discontinuous).

Plotting in geogebra shows the graphs of $ f_n $ getting ‘‘pinched”.