The answer to whether a rational function can have infinitely many x-values at which it is not continuous is;
False because it is only when the values of x make the denominator is zero, that the function becomes discontinuous.
A rational function is simply defined as the ratio of two polynomials. This means it is a division of 2 polynomials with one at the numerator and one at the denominator.
Now, the values of x for the polynomial can be any value that doesn't make the function undefined such as when the denominator is zero.
Undefined functions are discontinuous and therefore it is only when the value of x makes the denominator to be zero that the function becomes discontinuous.
Thus, in conclusion, since it is only the values of x that make the denominator to be zero that cause discontinuity, then we can say that the statement in the question is false.
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