If a function is continuous at a point, A. the limit of the function may not exist at the point B. the function must be derivable at the point C. the limit of the function at the point tends to infinity D. the limit must exist at the point and the value of limit should be same as the value of the function at that point

The correct answer is: D. The limit must exist at the point and the value of limit should be same as the value of the function at that point.

A function is continuous at a point if the following two conditions are met:

1. The function has a defined value at the point.
2. The limit of the function as $x$ approaches the point exists and is equal to the value of the function at the point.

In other words, if a function is continuous at a point, then the graph of the function must pass through that point and the two-sided limit of the function as $x$ approaches the point must be equal to the value of the function at the point.

Option A is incorrect because if a function is continuous at a point, then the limit of the function must exist at that point.

Option B is incorrect because a function can be continuous at a point without being derivable at that point. For example, the function $f(x) = |x|$ is continuous at $x = 0$, but it is not derivable at $x = 0$.

Option C is incorrect because the limit of a function at a point can be any finite number, including infinity. For example, the function $f(x) = \frac{1}{x}$ has a limit of infinity as $x$ approaches zero.

I hope this explanation is helpful!