Tell whether if the following piecewise function is a continuous at a given point or not. (SHOW THE SOLUTION).

Question

Tell whether if the following piecewise function is a continuous at a given point or not. (SHOW THE SOLUTION).

2. at x = 0

1/x if x ≤ -1

x² - 2 if x > -1

Answer 1

1/x is a rectangular hyperbola

for x<-1 it's part of the lower branch of the hyperbola

x^2 -2 is an upward opening parabola

the parabola and hyperboal meet at x=-1

the stepwise function is continuous at all points for all values of x including x=0

You might be tempted to compare the hyperbola to the parabola at x=0,

where the parabola is at its y intercept (0,-2) and the hyperboal is undefined, approaching negative infinity, but that ignores the fact that the stepwise function at x=0 is entirely continuous since only the parabola is relevant at x=0.