Original
To verify item {{CMD_125}} we must check that {{MATH_126}} exists. Again, the formula for {{MATH_127}} changes around {{MATH_128}} so we
check the one-sided limits. From the left:
{{ENV_129}}
Looking at the limit from the right, we have
{{ENV_130}}
In order for {{MATH_131}} to exist, these must be equal.
{{ENV_132}}
This means item {{CMD_133}} is satisfied for {{MATH_134}} and {{MATH_135}} with {{MATH_136}}.
Item {{CMD_137}} is then satisfied for all {{MATH_138}} and {{MATH_139}} with {{MATH_140}}, since this gives {{MATH_141}} and {{MATH_142}}.
So in order to be continuous at BOTH {{MATH_143}} AND {{MATH_144}}, we have two equations and two unknowns:
{{MATH_145}}
Set {{MATH_146}} and write
{{ENV_147}}
hence
{{MATH_148}}
Let's check, so now plugging in values for both {{MATH_149}} and {{MATH_150}} we find
{{MATH_151}}
Now
{{MATH_152}}
and
{{MATH_153}}
So setting {{MATH_154}} and {{MATH_155}} makes {{MATH_156}} continuous at {{MATH_157}} and {{MATH_158}}.
{{CMD_159}}
We can confirm our results by looking at the graph of {{MATH_160}}:
{{MATH_161}}
{{CMD_162}}
{{CMD_163}}
{{CMD_164}}