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Discuss the continuity of the following functions at the indicated point(s).
at x = 0
In this problem we need to check continuity at x = 0
Given function is
at x = 0 …… Equation 2
∴ we need to check LHL, RHL and value of function at x = 0 (for idea and meaning of continuity refer to Q10(i) )
f(0) = 0 [from equation 2]
LHL = [∵ sin(–θ) = sinθ]
RHL = [ using eqn 2]
Since whatever is value of h, sin(1/h) is going to range from –1 to 1
As h→ 0 i.e. h is approximately 0
∴ 0*some finite quantity is equal to 0.
Clearly, LHL = RHL = f(0)
∴ f(x) is continuous at x = 0