When functions are not continuous, limits and derivatives may be undefined. There are different kind of discontinuities.
Jump Discontinuities
It occurs when the right and left side limits exist but are not equal. See the following example.
Removable Discontinuities
The left-hand and right-hand limits are equal but either the function is not defined or not equal to these limits.
Infinite Discontinuities
The left-hand and right-hand limits are infinite
Other Discontinuities
Differentiable Implies Continuous
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