Final answer:
To make a function continuous everywhere, we need to ensure that there are no discontinuities or breaks in the function. This can be done by avoiding division by zero and ensuring that any operations with limited domain do not result in discontinuities.
Step-by-step explanation:
To make a function continuous everywhere, we need to ensure that there are no discontinuities or breaks in the function. In order to do this, we need to check for any values of a and b that would result in the function being discontinuous.
One common type of discontinuity is a removable discontinuity, which can occur when there is a point where the function is undefined, such as a division by zero. To avoid this, we need to make sure that the function is defined for all values of x, which means that the denominators of any fractions in the function cannot equal zero.
Additionally, if the function involves a square root, logarithm, or any other type of operation that is only defined for certain values of x, then we need to make sure that the values of x allowed by these operations do not result in discontinuities.
The values of a and b required to make a function continuous everywhere are determined by ensuring the function itself and its first derivative are continuous, except where potential values may be infinite. For a probability density function (pdf), the total area under the curve must equal one, indicating that the function accounts for the entire range of probable outcomes. Moreover, for continuous probability distributions, the probability of any single value, such as P(x=c), is zero, focusing instead on the probabilities within an interval, P(a ≤ x ≤ b).