1.suppose that f:X->Y is continuous. if x is a limit point of the subset A of X, is it necessarily true that f(x) is a limit point of f(A)?(adsbygoogle = window.adsbygoogle || []).push({});

2. suppose that f:R->R is continuous from the right, show that f is continuous when considered as a function from R_l to R, where R_l is R in the lower limit topology. (munkres' notation).

Now for 1, I think the answer is no, but I don't find a counterexample if someone could give me a hint on this ( I'm pretty sure it's easy (-: ), but what I did find is that if f is injective then the answer is yes (I proved it by ad absurdum), so my hunch a counterexample should be with a function which is not injective.

Now for two it seems easy enough, if V is open in R, then it contains an open interval, let it be (a,b), now then f^-1(V) contains f^-1((a,b))={x in R_l|f(x) in (a,b)}

now i need to prove that f^-1((a,b)) is an interval of the form: [x0,x1), but im struggling with that.

any hints?

thanks in advance.

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# Topological continuity (a few questions).

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