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While studying calculus, I've read about continuous functions but I still couldn't figure out what's the importance of the concept, I imagine that the concept (and also the concept of continuity) may have it's importance in calculus and also in some branches of higher mathematics - If I'm not mistaken, I've seen continuity in some book on topology.

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Continuity in calculus is more or less just a guarantee that the function behaves nicely in a few respects (especially concerning limits). –  Arthur Oct 3 '13 at 6:52

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One use for them is in applied mathematics when using numerical methods to approximate a value using Taylor's Theorem, which only works for $k^{th}$ order differentiable functions. If a function was discontinuous, Taylor's Theorem could fail.

See here for detailed explanation on Taylor's Theorem

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The importance of continuity is easiest explained by the Intermediate Value theorem : It says that, if a continuous function takes a positive value at one point, and a negative value at another point, then it must take the value zero somewhere in between.

One striking example of this fact is the following : Find the temperature at the south and north pole, and take their difference (call is $\alpha$). Then move from the north pole to the south pole, and at each point take the difference between the temperatures at that point and at the "diametrically opposite" point on the earth. When you finally reach the south pole, this difference will be $-\alpha$).

The Intermediate Value theorem now tells you that there must exist two diametrically opposite points on the earth with equal temperature.

Another "mathematical" example is that any polynomial of odd degree must have a (real) root; because as you go to $-\infty$, it must become negative, and as you go to $+\infty$, it must become positive eventually (Think about why this must happen)

Continuity is much more than these simple-minded examples, though, and you could look at some elementary books on topology/analysis to gain a deeper understanding.

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