A sequence of continuous functions, $\{f_n\}$ converges pointwise to a discontinuous function $g$, then $\{f_n\}$ does not converges uniformly to $g$?

Question

If a $\{f_n\}$ a sequence of continuous functions, converges pointwise to a discontinuous function $g$, does it imply that $\{f_n\}$ does not converges uniformly to $g$?

Answer 1

Suppose $\{f_n\}$ is a sequence of continuous functions such that $\{f_n\}$ converges pointwise to a discontinuous function $g$, then it must be the case that $\{f_n\}$ does not converge uniformly to $g$

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Proof:

If $\{f_n\}$ is a uniformly convergent sequence of continuous functions, then by definition $\forall \epsilon > 0, \exists N$ s.t. $\forall x \in M, \forall n \geq N, |f_n(x) - f(x)| < \epsilon$

So if $f(x)$ is discontinuous at a point, then there exists some $x_o \in M$ and $\epsilon > 0$ such that $|f_n(x_o) - f(x_o)| \geq \epsilon$

Hence $\{f_n\}$ must not be uniformly convergent.

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Example:

$f_n(x) = x^n$ on $[0,1]$

Each $f_n$ is continuous, and converges pointwise to $0, \forall x \in [0, 1)$ and $1, x =1$