Sequential Characterization of Continuity

Detailed Solution

The key idea is the sequential characterization of continuity. Suppose $$ x_n\to x. $$ If $f$ is continuous at $x$, then $$ f(x_n)\to f(x). $$ Thus continuity implies that every convergent sequence is mapped to a convergent sequence. We now show the converse. Assume that $f$ maps every convergent sequence to a convergent sequence. Let $$ x_n\to x. $$ We need to show that $$ f(x_n)\to f(x). $$ By the given condition, $f(x_n)$ converges to some limit, say $L$: $$ f(x_n)\to L. $$ Consider the interlaced sequence $$ y_1=x,\qquad y_2=x_1,\qquad y_3=x,\qquad y_4=x_2,\qquad\ldots $$ Equivalently, $$ y_{2n-1}=x,\qquad y_{2n}=x_n. $$ Since $x_n\to x$, we have $$ y_n\to x. $$ Therefore, by the assumed property, $$ f(y_n) $$ must converge. But $$ f(y_{2n-1})=f(x) $$ for every $n$, while $$ f(y_{2n})=f(x_n)\to L. $$ A convergent sequence and each of its subsequences must have the same limit. The odd subsequence is constantly equal to $f(x)$, so its limit is $f(x)$. The even subsequence has limit $L$. Hence $$ L=f(x). $$ Therefore, $$ f(x_n)\to f(x). $$ Since this holds for every sequence $x_n\to x$, $f$ is continuous at $x$. Since $x$ was arbitrary, $f$ is continuous on $\mathbb R$. Thus, $$ \boxed{\text{The given condition is equivalent to continuity of }f.} $$ Now consider the other options. A is false. Continuity does not imply uniform continuity on $\mathbb R$. For example, $$ f(x)=x^2 $$ is continuous on $\mathbb R$ but is not uniformly continuous on $\mathbb R$. B is weaker than C. The statement that $f$ is continuous is already assumed, whereas C identifies the given sequential property as an equivalent characterization of continuity. D is true for every continuous function on $\mathbb R$: if $[a,b]$ is a bounded closed interval, then $f$ is continuous on the compact set $[a,b]$, so it is bounded there by the Extreme Value Theorem. However, this is only a consequence of continuity and is weaker than the equivalence stated in C. Therefore, the strongest valid conclusion is $$ \boxed{\mathrm{C}}. $$