Is there a Tychonoff space $X$ without isolated points with the following property: 


For any $a\in X$ and any function $f : X\longrightarrow \mathbb{R}$, if $f$ is continuous on $X\backslash \{a\}$ then we can redefine $f$ at $x=a$ such that (the new) $f$ is continuous on $X$. 



Topological spaces with too many open sets
Consistently, $\omega^\*$ has this property. In the paper 



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it was shown to be consistent that, for every $p \in \omega^\*$, every **bounded** continuous real-valued function on $\omega^\* - \{p\}$ can be extended to $p$. This almost gives a consistent answer to the OP's question, the only problem being that the property in the question does not mention boundedness. However, in the context of $\omega^\*$ it turns out that adding in boundedness does not hurt anything:


**Observation:** *If $p \in \omega^\*$, then every real-valued continuous function on $\omega^\* - \{p\}$ is bounded.*


**Proof:** Suppose $f$ is an unbounded continuous real-valued function on $\omega^\* - \{p\}$. Pick a sequence $\langle x\_n : n < \omega \rangle$ of points in $\omega^\* - \{p\}$ such that $f(x\_n)$ converges to infinity. It is well known that $\omega^\*$ is compact and contains no nontrivial convergent sequences. Thus there must be a point $q \in \omega^\*$ with $q \neq p$ at which the $x\_n$ cluster. Since $f(q)$ is some (finite) real number but $f(x\_n)$ goes to infinity, this contradicts the continuity of $f$. $\qquad$ *QED*


On the other hand, Fine and Gillman proved that $\omega^\*$ consistently fails to have the OP's property (CH implies that it does not).


I do not know whether your question has a positive answer in ZFC (but I suspect that it does).


[Here](http://arxiv.org/pdf/1310.0678v3.pdf) is a link to a related paper that you might find interesting (it's where I learned about the two results I mention above).


