📝Continuity and Differentiability
Continuity at a point
●A real function `f(x)` is said to be continuous at a point `a` of its domain if ` \lim_{x\rightarrow a}f(x)` exists and equals `f(a)`.
Thus, `f(x)` is continuous at `x = a` if
` \lim_{x\rightarrow a+}f(x) = \lim_{x\rightarrow a-}f(x) = f(a)`
● If `f(x)` is not continuous at a point, it is said to be discontinuous at that point
Remarks `f(x)` is discontinuous at `x = a` in each of the following cases:
- `f(a)` is not defined
- ` \lim_{x\rightarrow a}f(x)` does not exist
- ` \lim_{x\rightarrow a}f(x) \ne f(a)`