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In metric spaces, we have this condition that a map $f$ is continuous at a point $x$ iff for each seq. $(X_n)$ converging to $x$ the seq. $(f(X_n))$ converges to $f(X)$. Is the same true in case of a topological space? i.e. is it true, A map $f$ on a topological space is continuous at a point $x$ iff for each net $(X_n)$ converging to $x$ the net $(f(X_n))$ converges to $f(x)$?

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It's not true for sequences. See this post. –  David Mitra Feb 1 '13 at 16:24
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See: [Elementary Topology][1], S 6.6, Proposition 10. [1]: books.google.com/… –  M.Sina Feb 1 '13 at 16:25

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Yes, this is true. This is more or less the point of nets. It's also true if you replace nets with filters.

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