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Extra info for Counter-examples in calculus
The function y x is continuous at the point x = 0 but it is not differentiable at that point. 57 y x 2 8 -6 -4 0 -2 0 2 4 6 8 -2 -4 4. If a function is continuous on R and the tangent line exists at any point on its graph then the function is differentiable at any point on R. Counter-example. 3 The function y x 2 is continuous on R and the tangent line exists at any point on its graph but the function is not differentiable at the point x = 0. 2 y 8 -6 -4 0 -2 0 2 4 3 x2 6 -2 -4 5. If a function is continuous on the interval (a,b) and its graph is a smooth curve (no sharp corners) on that interval then the function is differentiable at any point on (a,b).
Sx )) 2 n . k o f no f f ( x) 1 36 1, if x is rational ® ¯0, if x is irrational 2. Limits 1. If f(x) < g(x) for all x > 0 and both lim f ( x ) and lim g( x ) exist then xof xof lim f ( x ) lim g( x ) . x of x of Counter-example. For the functions f ( x ) but lim f ( x ) lim g( x ) x of xof 1 and g ( x ) x 1 x f(x) < g(x) for all x ! 0 0. 4 1 x g( x ) 2 -2 0 0 2 4 6 8 10 12 -2 f ( x) 2. The following equivalent: definitions of a non-vertical 1 x asymptote are a) The straight line y = mx + c is called a non-vertical asymptote to a curve f(x) as x tends to infinity if lim ( f ( x ) ( mx c )) 0 .
Takes its maximum and minimum values; c. takes all its values between the maximum and minimum values; then this function is continuous at some points or subintervals on [a,b]. Counter-example. The function below satisfies all three conditions above but it is discontinuous at every point on [-1,1]. It is impossible to draw the graph of the function y = f(x) but the sketch below gives an idea of its behaviour. 51 f ( x) 1, if x 0; ° x , if x is rational, x z 0, x z 1; ° ® ° x , if x is irrational, x z 0, x z 1, x z 1 °¯ 0, if x 1 1 -1 12.