Question

Find c, so that f(x) is everywhere continuous?

f(x)= {x+c, x <_2
2x^2 -5, x > 2

Answer 1

For f(x) to be continuous everywhere, it must be continuous in each of the two intervals (-∞2] and (2,∞), which is the case, since all polynomials are continuous.

In particular, the limit of f(x) as x approaches from 2- and 2+ must be also equal for f(x) to be continuous at x=2.

This requires that
x+c = 2x^2 -5 at x=2, or
2+c = 2(2)^2 - 5
c = 1