Continuity of a function $f:X \longrightarrow Y$ Where $X,Y$ metric space I know most of the result regarding Continuity, Completeness. But I got stuck into a question which I found in a question paper. Let $f:X \longrightarrow Y$ Where $X,Y$ metric space.Then show that for any Cauchy sequence $\{x_n\}$ in $X$ if $\{f(x_n)\}$ is Cauchy in $Y$, then $f$ is continuous. Please help me. Thanks in advance.
Is a Cauchy sequence - preserving (continuous) function is (uniformly) continuous? Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$ be a function and suppose for any Cauchy sequence $(a_n)$ in $X$, $(f(a_n))$ is a Cauchy sequence in $Y$.  Is $f$ continuous?  Let $f$ be continuous, is it uniformly continuous?
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The courses I have completed are basically both the undergraduate and graduate-level algebra sequence,  analysis in R and R^n (without a formal introduction to Lebesgue measure), discrete math (for EECS), applied probability, one course in numerical methods, PDEs, applied complex variables, and a "mathematical methods for sciences" course (focusing on special functions, variational calculus, and integral transforms).   I won't have the standard math major courses(i.e,  topology, measure theory, and number theory.) I plan to focus more on applications than theory in my combinatorics Ph.D.   Would additional coursework be necessary for a Ph.D. in combinatorics (specifically graph theory)? (I could potentially delay my degree by a year and spend time as a non-degree student at another university, or complete a non-thesis track master's degree before applying for a masters/Ph.d.). Any advice would be appreciated. How does finder sort folders when they contain digits and characters? 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