For what values of a and b is this function continuous

So there's a piecewise function,

f(x) = a(√x) + b , x>0
a , x=0
2a - bx² , x<0

For what values of a, b is this function continuous at 0?

Since the function is continuous if and only if the limit of f(x) as x approaches zero exists and equals the value of the function at this point, I got that 2a = b = a, which implies the function is only continuous at 0 if a, b=0. But........... that means that the whole function is 0 (ie. f(x) = 0 for all values of x)! What did I do wrong? If this is correct, is it correct to say a, b = 0, or should I say it can't be continuous at x=0?

I will appreciate any help with this, I am quite perplexed! Thank you!

Everything you did was correct. It is weird that they would give you a problem where the whole function is equal to 0. Double check the original problem (I always seem to write the problem down wrong.). It would be correct to say that the function is only continuous is a,b=0.

The function f(x) = a(√x) + b , x>0

Is not continous at x= 0 ... because a function to be continous at x=0, must have image at x=0 and must have limt (x->0) and the limit must be the same as the image.
But here as x>0, then there is no image at x=0, then f(x) is not continuous for any value of a and b.
OK!