challenge someone on something
to dispute someone's statement, remarks, or position. I think Fred is wrong, but I won't challenge him on his estimate.
challenge someone to something
to dare someone to do something; to invite someone to compete at something. I challenge you to a round of golf. Jerry challenged us to a debate of the issues.
rise to the challenge
Fig. to accept a challenge. (Usually in reference to success with the challenge.) You can depend on Kelly to rise to the challenge. We were not able to rise to the challenge and we lost the contract.
take up the challenge
to respond to a challenge and do what the challenge asks. I am not prepared to take the challenge up. Dave took up the challenge without much urging.