What makes a function continuous at a point?

Hint: We shall define the continuity of a function by using the concept of limits. A function $f\left( x \right)$ is said to be continuous at a point $x=b$ if the function exhibits continuity at this point and there are no discontinuities in the path of the function at this point. We then define the same in terms of limits and explain the continuity concept.

Complete step-by-step solution:
In order to answer this question, let us first define the concept of continuity in terms of limits. The continuity of a function $f\left( x \right)$ at a point $x=b$ can be determined based on the function values at these points. We need to take the left-hand limit and right-hand limit for the point $x=b$ and calculate the values to check if they are equal and equal to the value of the function at this point. If these two limits are equal, then we can say that the function is continuous at this point which means that there exists a value at every small interval in the function and there are no breaks or discontinuities in this function.
We can represent the same in the mathematical form using equations. To find out whether a function $f\left( x \right)$ is continuous at a point $x=b$ , we consider the two limits for the function. They are the left-hand limit represented by ${{b}^{-}}$ and the right-hand limit represented by ${{b}^{+}}.$
If, $\displaystyle \lim_{x \to {{b}^{-}}}f\left( x \right)=\displaystyle \lim_{x \to {{b}^{+}}}f\left( x \right)=f\left( b \right),$ then we can say that the function is continuous at a point.
Hence, we have explained the concept of a continuous function.

Note: We need to note that the differentiability of a function is related to its continuity. If a function is differentiable at a point, then it is continuous at the point. The reverse need not be true. If a function is continuous at a point, then it may or may not be differentiable at this point.