So, the sequential criteria for continuity given in Bartle and Sherbert is that, a function $f:A\mapsto R$ is continuous at point $c\in A$ if and only if for every sequence $(x_n)$ in $A$ that converges to $c$, the sequence $(f(x_n))$ converges to $f(c)$.
And the sequential criteria for uniform continuity is : Let $f:A\mapsto R$. If $x_n$ and $y_n$ be sequences on $A$ such that $lim_{n\to\infty}(x_n)=lim_{n\to\infty}(y_n)$, then $f$ is uniformly continuous on A $\iff$ $lim_{n\to\infty}(f(x_n))=lim_{n\to\infty}(f(y_n))$
So, my doubt is if for some function, $lim_{n\to\infty}(f(x_n))=lim_{n\to\infty}(f(y_n))$=$\infty$, then will the function be uniformly continuous?
Similarly if $f(c)=\infty$ and $lim_{n\to\infty}(f(x_n))=\infty$, will the said function be continuous at $c$?
Since, the co-domain is $\mathbb{R}$ and $\infty \not \in \mathbb{R}$ assigning infinity as the value of $f$ at $c$ is invalid. Usually in such cases we just say that the function is undefined at $c$ (I am assuming that when you wrote $f(x)=c$ you meant that the function is indeterminate).
However if the co-domain has $\infty$ as an element(like the extended real numbers) it is valid to say $f(c) = \infty$ and in such a case, the answer to your question is yes.