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Continuity of piecewise function
x=1
Left limit = 2 × 1 + 1 = 3
Limit = 1-1 + 3 = 3
continuity
x=2
Left limit = 4-2 + 3 = 5
Limit = 8-1 = 7
Discontinuity
How to prove the continuity and differentiability of a function at a certain point?
First, prove with infinity that the infinity on the left has a value at this point, and then prove that the infinity on the right has a value. Then the two values are equal. Its function image must be continuous
The problem of proving the continuity of function
It's the problem in the picture. Don't use the proof method in calculus. It's better to use the original image of F continuous open set to prove it
x. In what scope is y discussed? I'll treat it as a topological group ~ it should be broad enough
According to the definition of continuity in topology, the original image of open set is open set
H (x, y) = u (x, y ^ (- 1)) = u (x, V (y)). Since u and V are continuous, h is continuous
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